Polynomials Manipulation Module Reference

Basic polynomial manipulation functions

sympy.polys.polytools.poly(expr, *gens, **args)

Efficiently transform an expression into a polynomial.

Examples

>>> from sympy import poly
>>> from sympy.abc import x

>>> poly(x*(x**2 + x - 1)**2)
Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')

sympy.polys.polytools.poly_from_expr(expr, *gens, **args)

Construct a polynomial from an expression.

sympy.polys.polytools.parallel_poly_from_expr(exprs, *gens, **args)

Construct polynomials from expressions.

sympy.polys.polytools.degree(f, gen=0)

Return the degree of f in the given variable.

The degree of 0 is negative infinity.

Examples

>>> from sympy import degree
>>> from sympy.abc import x, y

>>> degree(x**2 + y*x + 1, gen=x)
2
>>> degree(x**2 + y*x + 1, gen=y)
1
>>> degree(0, x)
-oo

See also

total_degree, degree_list

sympy.polys.polytools.degree_list(f, *gens, **args)

Return a list of degrees of f in all variables.

Examples

>>> from sympy import degree_list
>>> from sympy.abc import x, y

>>> degree_list(x**2 + y*x + 1)
(2, 1)

sympy.polys.polytools.LC(f, *gens, **args)

Return the leading coefficient of f.

Examples

>>> from sympy import LC
>>> from sympy.abc import x, y

>>> LC(4*x**2 + 2*x*y**2 + x*y + 3*y)
4

sympy.polys.polytools.LM(f, *gens, **args)

Return the leading monomial of f.

Examples

>>> from sympy import LM
>>> from sympy.abc import x, y

>>> LM(4*x**2 + 2*x*y**2 + x*y + 3*y)
x**2

sympy.polys.polytools.LT(f, *gens, **args)

Return the leading term of f.

Examples

>>> from sympy import LT
>>> from sympy.abc import x, y

>>> LT(4*x**2 + 2*x*y**2 + x*y + 3*y)
4*x**2

sympy.polys.polytools.pdiv(f, g, *gens, **args)

Compute polynomial pseudo-division of f and g.

Examples

>>> from sympy import pdiv
>>> from sympy.abc import x

>>> pdiv(x**2 + 1, 2*x - 4)
(2*x + 4, 20)

sympy.polys.polytools.prem(f, g, *gens, **args)

Compute polynomial pseudo-remainder of f and g.

Examples

>>> from sympy import prem
>>> from sympy.abc import x

>>> prem(x**2 + 1, 2*x - 4)
20

sympy.polys.polytools.pquo(f, g, *gens, **args)

Compute polynomial pseudo-quotient of f and g.

Examples

>>> from sympy import pquo
>>> from sympy.abc import x

>>> pquo(x**2 + 1, 2*x - 4)
2*x + 4
>>> pquo(x**2 - 1, 2*x - 1)
2*x + 1

sympy.polys.polytools.pexquo(f, g, *gens, **args)

Compute polynomial exact pseudo-quotient of f and g.

Examples

>>> from sympy import pexquo
>>> from sympy.abc import x

>>> pexquo(x**2 - 1, 2*x - 2)
2*x + 2

>>> pexquo(x**2 + 1, 2*x - 4)
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1

sympy.polys.polytools.div(f, g, *gens, **args)

Compute polynomial division of f and g.

Examples

>>> from sympy import div, ZZ, QQ
>>> from sympy.abc import x

>>> div(x**2 + 1, 2*x - 4, domain=ZZ)
(0, x**2 + 1)
>>> div(x**2 + 1, 2*x - 4, domain=QQ)
(x/2 + 1, 5)

sympy.polys.polytools.rem(f, g, *gens, **args)

Compute polynomial remainder of f and g.

Examples

>>> from sympy import rem, ZZ, QQ
>>> from sympy.abc import x

>>> rem(x**2 + 1, 2*x - 4, domain=ZZ)
x**2 + 1
>>> rem(x**2 + 1, 2*x - 4, domain=QQ)
5

sympy.polys.polytools.quo(f, g, *gens, **args)

Compute polynomial quotient of f and g.

Examples

>>> from sympy import quo
>>> from sympy.abc import x

>>> quo(x**2 + 1, 2*x - 4)
x/2 + 1
>>> quo(x**2 - 1, x - 1)
x + 1

sympy.polys.polytools.exquo(f, g, *gens, **args)

Compute polynomial exact quotient of f and g.

Examples

>>> from sympy import exquo
>>> from sympy.abc import x

>>> exquo(x**2 - 1, x - 1)
x + 1

>>> exquo(x**2 + 1, 2*x - 4)
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1

sympy.polys.polytools.half_gcdex(f, g, *gens, **args)

Half extended Euclidean algorithm of f and g.

Returns (s, h) such that h = gcd(f, g) and s*f = h (mod g).

Examples

>>> from sympy import half_gcdex
>>> from sympy.abc import x

>>> half_gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)
(3/5 - x/5, x + 1)

sympy.polys.polytools.gcdex(f, g, *gens, **args)

Extended Euclidean algorithm of f and g.

Returns (s, t, h) such that h = gcd(f, g) and s*f + t*g = h.

Examples

>>> from sympy import gcdex
>>> from sympy.abc import x

>>> gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)
(3/5 - x/5, x**2/5 - 6*x/5 + 2, x + 1)

sympy.polys.polytools.invert(f, g, *gens, **args)

Invert f modulo g when possible.

Examples

>>> from sympy import invert, S
>>> from sympy.core.numbers import mod_inverse
>>> from sympy.abc import x

>>> invert(x**2 - 1, 2*x - 1)
-4/3

>>> invert(x**2 - 1, x - 1)
Traceback (most recent call last):
...
NotInvertible: zero divisor

For more efficient inversion of Rationals, use the mod_inverse function:

>>> mod_inverse(3, 5)
2
>>> (S(2)/5).invert(S(7)/3)
5/2

See also

sympy.core.numbers.mod_inverse

sympy.polys.polytools.subresultants(f, g, *gens, **args)

Compute subresultant PRS of f and g.

Examples

>>> from sympy import subresultants
>>> from sympy.abc import x

>>> subresultants(x**2 + 1, x**2 - 1)
[x**2 + 1, x**2 - 1, -2]

sympy.polys.polytools.resultant(f, g, *gens, **args)

Compute resultant of f and g.

Examples

>>> from sympy import resultant
>>> from sympy.abc import x

>>> resultant(x**2 + 1, x**2 - 1)
4

sympy.polys.polytools.discriminant(f, *gens, **args)

Compute discriminant of f.

Examples

>>> from sympy import discriminant
>>> from sympy.abc import x

>>> discriminant(x**2 + 2*x + 3)
-8

sympy.polys.dispersion.dispersion(p, q=None, *gens, **args)

Compute the dispersion of polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion \(\operatorname{dis}(f, g)\) is defined as:

\[\begin{split}\operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \}\end{split}\]

and for a single polynomial \(\operatorname{dis}(f) := \operatorname{dis}(f, f)\). Note that we make the definition \(\max\{\} := -\infty\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

The maximum of an empty set is defined to be \(-\infty\) as seen in this example.

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersionset

References

R549

[ManWright94]

R550

[Koepf98]

R551

[Abramov71]

R552

[Man93]

sympy.polys.dispersion.dispersionset(p, q=None, *gens, **args)

Compute the dispersion set of two polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion set \(\operatorname{J}(f, g)\) is defined as:

\[\begin{split}\operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}\end{split}\]

For a single polynomial one defines \(\operatorname{J}(f) := \operatorname{J}(f, f)\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersion

References

R553

[ManWright94]

R554

[Koepf98]

R555

[Abramov71]

R556

[Man93]

sympy.polys.polytools.terms_gcd(f, *gens, **args)

Remove GCD of terms from f.

If the deep flag is True, then the arguments of f will have terms_gcd applied to them.

If a fraction is factored out of f and f is an Add, then an unevaluated Mul will be returned so that automatic simplification does not redistribute it. The hint clear, when set to False, can be used to prevent such factoring when all coefficients are not fractions.

Examples

>>> from sympy import terms_gcd, cos
>>> from sympy.abc import x, y
>>> terms_gcd(x**6*y**2 + x**3*y, x, y)
x**3*y*(x**3*y + 1)

The default action of polys routines is to expand the expression given to them. terms_gcd follows this behavior:

>>> terms_gcd((3+3*x)*(x+x*y))
3*x*(x*y + x + y + 1)

If this is not desired then the hint expand can be set to False. In this case the expression will be treated as though it were comprised of one or more terms:

>>> terms_gcd((3+3*x)*(x+x*y), expand=False)
(3*x + 3)*(x*y + x)

In order to traverse factors of a Mul or the arguments of other functions, the deep hint can be used:

>>> terms_gcd((3 + 3*x)*(x + x*y), expand=False, deep=True)
3*x*(x + 1)*(y + 1)
>>> terms_gcd(cos(x + x*y), deep=True)
cos(x*(y + 1))

Rationals are factored out by default:

>>> terms_gcd(x + y/2)
(2*x + y)/2

Only the y-term had a coefficient that was a fraction; if one does not want to factor out the 1/2 in cases like this, the flag clear can be set to False:

>>> terms_gcd(x + y/2, clear=False)
x + y/2
>>> terms_gcd(x*y/2 + y**2, clear=False)
y*(x/2 + y)

The clear flag is ignored if all coefficients are fractions:

>>> terms_gcd(x/3 + y/2, clear=False)
(2*x + 3*y)/6

See also

sympy.core.exprtools.gcd_terms, sympy.core.exprtools.factor_terms

sympy.polys.polytools.cofactors(f, g, *gens, **args)

Compute GCD and cofactors of f and g.

Returns polynomials (h, cff, cfg) such that h = gcd(f, g), and cff = quo(f, h) and cfg = quo(g, h) are, so called, cofactors of f and g.

Examples

>>> from sympy import cofactors
>>> from sympy.abc import x

>>> cofactors(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)

sympy.polys.polytools.gcd(f, g=None, *gens, **args)

Compute GCD of f and g.

Examples

>>> from sympy import gcd
>>> from sympy.abc import x

>>> gcd(x**2 - 1, x**2 - 3*x + 2)
x - 1

sympy.polys.polytools.gcd_list(seq, *gens, **args)

Compute GCD of a list of polynomials.

Examples

>>> from sympy import gcd_list
>>> from sympy.abc import x

>>> gcd_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])
x - 1

sympy.polys.polytools.lcm(f, g=None, *gens, **args)

Compute LCM of f and g.

Examples

>>> from sympy import lcm
>>> from sympy.abc import x

>>> lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2

sympy.polys.polytools.lcm_list(seq, *gens, **args)

Compute LCM of a list of polynomials.

Examples

>>> from sympy import lcm_list
>>> from sympy.abc import x

>>> lcm_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])
x**5 - x**4 - 2*x**3 - x**2 + x + 2

sympy.polys.polytools.trunc(f, p, *gens, **args)

Reduce f modulo a constant p.

Examples

>>> from sympy import trunc
>>> from sympy.abc import x

>>> trunc(2*x**3 + 3*x**2 + 5*x + 7, 3)
-x**3 - x + 1

sympy.polys.polytools.monic(f, *gens, **args)

Divide all coefficients of f by LC(f).

Examples

>>> from sympy import monic
>>> from sympy.abc import x

>>> monic(3*x**2 + 4*x + 2)
x**2 + 4*x/3 + 2/3

sympy.polys.polytools.content(f, *gens, **args)

Compute GCD of coefficients of f.

Examples

>>> from sympy import content
>>> from sympy.abc import x

>>> content(6*x**2 + 8*x + 12)
2

sympy.polys.polytools.primitive(f, *gens, **args)

Compute content and the primitive form of f.

Examples

>>> from sympy.polys.polytools import primitive
>>> from sympy.abc import x

>>> primitive(6*x**2 + 8*x + 12)
(2, 3*x**2 + 4*x + 6)

>>> eq = (2 + 2*x)*x + 2

Expansion is performed by default:

>>> primitive(eq)
(2, x**2 + x + 1)

Set expand to False to shut this off. Note that the extraction will not be recursive; use the as_content_primitive method for recursive, non-destructive Rational extraction.

>>> primitive(eq, expand=False)
(1, x*(2*x + 2) + 2)

>>> eq.as_content_primitive()
(2, x*(x + 1) + 1)

sympy.polys.polytools.compose(f, g, *gens, **args)

Compute functional composition f(g).

Examples

>>> from sympy import compose
>>> from sympy.abc import x

>>> compose(x**2 + x, x - 1)
x**2 - x

sympy.polys.polytools.decompose(f, *gens, **args)

Compute functional decomposition of f.

Examples

>>> from sympy import decompose
>>> from sympy.abc import x

>>> decompose(x**4 + 2*x**3 - x - 1)
[x**2 - x - 1, x**2 + x]

sympy.polys.polytools.sturm(f, *gens, **args)

Compute Sturm sequence of f.

Examples

>>> from sympy import sturm
>>> from sympy.abc import x

>>> sturm(x**3 - 2*x**2 + x - 3)
[x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4]

sympy.polys.polytools.gff_list(f, *gens, **args)

Compute a list of greatest factorial factors of f.

Note that the input to ff() and rf() should be Poly instances to use the definitions here.

Examples

>>> from sympy import gff_list, ff, Poly
>>> from sympy.abc import x

>>> f = Poly(x**5 + 2*x**4 - x**3 - 2*x**2, x)

>>> gff_list(f)
[(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]

>>> (ff(Poly(x), 1)*ff(Poly(x + 2), 4)).expand() == f
True

>>> f = Poly(x**12 + 6*x**11 - 11*x**10 - 56*x**9 + 220*x**8 + 208*x**7 -         1401*x**6 + 1090*x**5 + 2715*x**4 - 6720*x**3 - 1092*x**2 + 5040*x, x)

>>> gff_list(f)
[(Poly(x**3 + 7, x, domain='ZZ'), 2), (Poly(x**2 + 5*x, x, domain='ZZ'), 3)]

>>> ff(Poly(x**3 + 7, x), 2)*ff(Poly(x**2 + 5*x, x), 3) == f
True

sympy.polys.polytools.gff(f, *gens, **args)

Compute greatest factorial factorization of f.

sympy.polys.polytools.sqf_norm(f, *gens, **args)

Compute square-free norm of f.

Returns s, f, r, such that g(x) = f(x-sa) and r(x) = Norm(g(x)) is a square-free polynomial over K, where a is the algebraic extension of the ground domain.

Examples

>>> from sympy import sqf_norm, sqrt
>>> from sympy.abc import x

>>> sqf_norm(x**2 + 1, extension=[sqrt(3)])
(1, x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16)

sympy.polys.polytools.sqf_part(f, *gens, **args)

Compute square-free part of f.

Examples

>>> from sympy import sqf_part
>>> from sympy.abc import x

>>> sqf_part(x**3 - 3*x - 2)
x**2 - x - 2

sympy.polys.polytools.sqf_list(f, *gens, **args)

Compute a list of square-free factors of f.

Examples

>>> from sympy import sqf_list
>>> from sympy.abc import x

>>> sqf_list(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)
(2, [(x + 1, 2), (x + 2, 3)])

sympy.polys.polytools.sqf(f, *gens, **args)

Compute square-free factorization of f.

Examples

>>> from sympy import sqf
>>> from sympy.abc import x

>>> sqf(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)
2*(x + 1)**2*(x + 2)**3

sympy.polys.polytools.factor_list(f, *gens, **args)

Compute a list of irreducible factors of f.

Examples

>>> from sympy import factor_list
>>> from sympy.abc import x, y

>>> factor_list(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)
(2, [(x + y, 1), (x**2 + 1, 2)])

sympy.polys.polytools.factor(f, *gens, **args)

Compute the factorization of expression, f, into irreducibles. (To factor an integer into primes, use factorint.)

There two modes implemented: symbolic and formal. If f is not an instance of Poly and generators are not specified, then the former mode is used. Otherwise, the formal mode is used.

In symbolic mode, factor() will traverse the expression tree and factor its components without any prior expansion, unless an instance of Add is encountered (in this case formal factorization is used). This way factor() can handle large or symbolic exponents.

By default, the factorization is computed over the rationals. To factor over other domain, e.g. an algebraic or finite field, use appropriate options: extension, modulus or domain.

Examples

>>> from sympy import factor, sqrt, exp
>>> from sympy.abc import x, y

>>> factor(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)
2*(x + y)*(x**2 + 1)**2

>>> factor(x**2 + 1)
x**2 + 1
>>> factor(x**2 + 1, modulus=2)
(x + 1)**2
>>> factor(x**2 + 1, gaussian=True)
(x - I)*(x + I)

>>> factor(x**2 - 2, extension=sqrt(2))
(x - sqrt(2))*(x + sqrt(2))

>>> factor((x**2 - 1)/(x**2 + 4*x + 4))
(x - 1)*(x + 1)/(x + 2)**2
>>> factor((x**2 + 4*x + 4)**10000000*(x**2 + 1))
(x + 2)**20000000*(x**2 + 1)

By default, factor deals with an expression as a whole:

>>> eq = 2**(x**2 + 2*x + 1)
>>> factor(eq)
2**(x**2 + 2*x + 1)

If the deep flag is True then subexpressions will be factored:

>>> factor(eq, deep=True)
2**((x + 1)**2)

If the fraction flag is False then rational expressions won’t be combined. By default it is True.

>>> factor(5*x + 3*exp(2 - 7*x), deep=True)
(5*x*exp(7*x) + 3*exp(2))*exp(-7*x)
>>> factor(5*x + 3*exp(2 - 7*x), deep=True, fraction=False)
5*x + 3*exp(2)*exp(-7*x)

See also

sympy.ntheory.factor_.factorint

sympy.polys.polytools.intervals(F, all=False, eps=None, inf=None, sup=None, strict=False, fast=False, sqf=False)

Compute isolating intervals for roots of f.

Examples

>>> from sympy import intervals
>>> from sympy.abc import x

>>> intervals(x**2 - 3)
[((-2, -1), 1), ((1, 2), 1)]
>>> intervals(x**2 - 3, eps=1e-2)
[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]

sympy.polys.polytools.refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False)

Refine an isolating interval of a root to the given precision.

Examples

>>> from sympy import refine_root
>>> from sympy.abc import x

>>> refine_root(x**2 - 3, 1, 2, eps=1e-2)
(19/11, 26/15)

sympy.polys.polytools.count_roots(f, inf=None, sup=None)

Return the number of roots of f in [inf, sup] interval.

If one of inf or sup is complex, it will return the number of roots in the complex rectangle with corners at inf and sup.

Examples

>>> from sympy import count_roots, I
>>> from sympy.abc import x

>>> count_roots(x**4 - 4, -3, 3)
2
>>> count_roots(x**4 - 4, 0, 1 + 3*I)
1

sympy.polys.polytools.real_roots(f, multiple=True)

Return a list of real roots with multiplicities of f.

Examples

>>> from sympy import real_roots
>>> from sympy.abc import x

>>> real_roots(2*x**3 - 7*x**2 + 4*x + 4)
[-1/2, 2, 2]

sympy.polys.polytools.nroots(f, n=15, maxsteps=50, cleanup=True)

Compute numerical approximations of roots of f.

Examples

>>> from sympy import nroots
>>> from sympy.abc import x

>>> nroots(x**2 - 3, n=15)
[-1.73205080756888, 1.73205080756888]
>>> nroots(x**2 - 3, n=30)
[-1.73205080756887729352744634151, 1.73205080756887729352744634151]

sympy.polys.polytools.ground_roots(f, *gens, **args)

Compute roots of f by factorization in the ground domain.

Examples

>>> from sympy import ground_roots
>>> from sympy.abc import x

>>> ground_roots(x**6 - 4*x**4 + 4*x**3 - x**2)
{0: 2, 1: 2}

sympy.polys.polytools.nth_power_roots_poly(f, n, *gens, **args)

Construct a polynomial with n-th powers of roots of f.

Examples

>>> from sympy import nth_power_roots_poly, factor, roots
>>> from sympy.abc import x

>>> f = x**4 - x**2 + 1
>>> g = factor(nth_power_roots_poly(f, 2))

>>> g
(x**2 - x + 1)**2

>>> R_f = [ (r**2).expand() for r in roots(f) ]
>>> R_g = roots(g).keys()

>>> set(R_f) == set(R_g)
True

sympy.polys.polytools.cancel(f, *gens, **args)

Cancel common factors in a rational function f.

Examples

>>> from sympy import cancel, sqrt, Symbol
>>> from sympy.abc import x
>>> A = Symbol('A', commutative=False)

>>> cancel((2*x**2 - 2)/(x**2 - 2*x + 1))
(2*x + 2)/(x - 1)
>>> cancel((sqrt(3) + sqrt(15)*A)/(sqrt(2) + sqrt(10)*A))
sqrt(6)/2

sympy.polys.polytools.reduced(f, G, *gens, **args)

Reduces a polynomial f modulo a set of polynomials G.

Given a polynomial f and a set of polynomials G = (g_1, ..., g_n), computes a set of quotients q = (q_1, ..., q_n) and the remainder r such that f = q_1*g_1 + ... + q_n*g_n + r, where r vanishes or r is a completely reduced polynomial with respect to G.

Examples

>>> from sympy import reduced
>>> from sympy.abc import x, y

>>> reduced(2*x**4 + y**2 - x**2 + y**3, [x**3 - x, y**3 - y])
([2*x, 1], x**2 + y**2 + y)

sympy.polys.polytools.groebner(F, *gens, **args)

Computes the reduced Groebner basis for a set of polynomials.

Use the order argument to set the monomial ordering that will be used to compute the basis. Allowed orders are lex, grlex and grevlex. If no order is specified, it defaults to lex.

For more information on Groebner bases, see the references and the docstring of \(solve_poly_system()\).

Examples

Example taken from [1].

>>> from sympy import groebner
>>> from sympy.abc import x, y

>>> F = [x*y - 2*y, 2*y**2 - x**2]

>>> groebner(F, x, y, order='lex')
GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y,
              domain='ZZ', order='lex')
>>> groebner(F, x, y, order='grlex')
GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,
              domain='ZZ', order='grlex')
>>> groebner(F, x, y, order='grevlex')
GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,
              domain='ZZ', order='grevlex')

By default, an improved implementation of the Buchberger algorithm is used. Optionally, an implementation of the F5B algorithm can be used. The algorithm can be set using method flag or with the setup() function from sympy.polys.polyconfig:

>>> F = [x**2 - x - 1, (2*x - 1) * y - (x**10 - (1 - x)**10)]

>>> groebner(F, x, y, method='buchberger')
GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex')
>>> groebner(F, x, y, method='f5b')
GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex')

References

1. [Buchberger01]

2. [Cox97]

sympy.polys.polytools.is_zero_dimensional(F, *gens, **args)

Checks if the ideal generated by a Groebner basis is zero-dimensional.

The algorithm checks if the set of monomials not divisible by the leading monomial of any element of F is bounded.

References

David A. Cox, John B. Little, Donal O’Shea. Ideals, Varieties and Algorithms, 3rd edition, p. 230

class sympy.polys.polytools.Poly

Generic class for representing and operating on polynomial expressions. Subclasses Expr class.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

Create a univariate polynomial:

>>> Poly(x*(x**2 + x - 1)**2)
Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')

Create a univariate polynomial with specific domain:

>>> from sympy import sqrt
>>> Poly(x**2 + 2*x + sqrt(3), domain='R')
Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR')

Create a multivariate polynomial:

>>> Poly(y*x**2 + x*y + 1)
Poly(x**2*y + x*y + 1, x, y, domain='ZZ')

Create a univariate polynomial, where y is a constant:

>>> Poly(y*x**2 + x*y + 1,x)
Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]')

You can evaluate the above polynomial as a function of y:

>>> Poly(y*x**2 + x*y + 1,x).eval(2)
6*y + 1

See also

sympy.core.expr.Expr

EC(order=None)

Returns the last non-zero coefficient of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x**2 + 3*x, x).EC()
3

EM(order=None)

Returns the last non-zero monomial of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM()
x**0*y**1

ET(order=None)

Returns the last non-zero term of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET()
(x**0*y**1, 3)

LC(order=None)

Returns the leading coefficient of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC()
4

LM(order=None)

Returns the leading monomial of f.

The Leading monomial signifies the monomial having the highest power of the principal generator in the expression f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM()
x**2*y**0

LT(order=None)

Returns the leading term of f.

The Leading term signifies the term having the highest power of the principal generator in the expression f along with its coefficient.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT()
(x**2*y**0, 4)

TC()

Returns the trailing coefficient of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x**2 + 3*x, x).TC()
0

abs()

Make all coefficients in f positive.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).abs()
Poly(x**2 + 1, x, domain='ZZ')

add(g)

Add two polynomials f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).add(Poly(x - 2, x))
Poly(x**2 + x - 1, x, domain='ZZ')

>>> Poly(x**2 + 1, x) + Poly(x - 2, x)
Poly(x**2 + x - 1, x, domain='ZZ')

add_ground(coeff)

Add an element of the ground domain to f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x + 1).add_ground(2)
Poly(x + 3, x, domain='ZZ')

all_coeffs()

Returns all coefficients from a univariate polynomial f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x - 1, x).all_coeffs()
[1, 0, 2, -1]

all_monoms()

Returns all monomials from a univariate polynomial f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x - 1, x).all_monoms()
[(3,), (2,), (1,), (0,)]

See also

all_terms

all_roots(multiple=True, radicals=True)

Return a list of real and complex roots with multiplicities.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).all_roots()
[CRootOf(x**3 + x + 1, 0),
 CRootOf(x**3 + x + 1, 1),
 CRootOf(x**3 + x + 1, 2)]

all_terms()

Returns all terms from a univariate polynomial f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x - 1, x).all_terms()
[((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]

args

Don’t mess up with the core.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).args
(x**2 + 1,)

as_dict(native=False, zero=False)

Switch to a dict representation.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict()
{(0, 1): -1, (1, 2): 2, (2, 0): 1}

as_expr(*gens)

Convert a Poly instance to an Expr instance.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = Poly(x**2 + 2*x*y**2 - y, x, y)

>>> f.as_expr()
x**2 + 2*x*y**2 - y
>>> f.as_expr({x: 5})
10*y**2 - y + 25
>>> f.as_expr(5, 6)
379

as_list(native=False)

Switch to a list representation.

cancel(g, include=False)

Cancel common factors in a rational function f/g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x))
(1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))

>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True)
(Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))

clear_denoms(convert=False)

Clear denominators, but keep the ground domain.

Examples

>>> from sympy import Poly, S, QQ
>>> from sympy.abc import x

>>> f = Poly(x/2 + S(1)/3, x, domain=QQ)

>>> f.clear_denoms()
(6, Poly(3*x + 2, x, domain='QQ'))
>>> f.clear_denoms(convert=True)
(6, Poly(3*x + 2, x, domain='ZZ'))

coeff_monomial(monom)

Returns the coefficient of monom in f if there, else None.

Examples

>>> from sympy import Poly, exp
>>> from sympy.abc import x, y

>>> p = Poly(24*x*y*exp(8) + 23*x, x, y)

>>> p.coeff_monomial(x)
23
>>> p.coeff_monomial(y)
0
>>> p.coeff_monomial(x*y)
24*exp(8)

Note that Expr.coeff() behaves differently, collecting terms if possible; the Poly must be converted to an Expr to use that method, however:

>>> p.as_expr().coeff(x)
24*y*exp(8) + 23
>>> p.as_expr().coeff(y)
24*x*exp(8)
>>> p.as_expr().coeff(x*y)
24*exp(8)

See also

nth

more efficient query using exponents of the monomial’s generators

coeffs(order=None)

Returns all non-zero coefficients from f in lex order.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 + 2*x + 3, x).coeffs()
[1, 2, 3]

See also

all_coeffs, coeff_monomial, nth

cofactors(g)

Returns the GCD of f and g and their cofactors.

Returns polynomials (h, cff, cfg) such that h = gcd(f, g), and cff = quo(f, h) and cfg = quo(g, h) are, so called, cofactors of f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x))
(Poly(x - 1, x, domain='ZZ'),
 Poly(x + 1, x, domain='ZZ'),
 Poly(x - 2, x, domain='ZZ'))

compose(g)

Computes the functional composition of f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + x, x).compose(Poly(x - 1, x))
Poly(x**2 - x, x, domain='ZZ')

content()

Returns the GCD of polynomial coefficients.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(6*x**2 + 8*x + 12, x).content()
2

count_roots(inf=None, sup=None)

Return the number of roots of f in [inf, sup] interval.

Examples

>>> from sympy import Poly, I
>>> from sympy.abc import x

>>> Poly(x**4 - 4, x).count_roots(-3, 3)
2
>>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I)
1

decompose()

Computes a functional decomposition of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose()
[Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')]

deflate()

Reduce degree of f by mapping x_i**m to y_i.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate()
((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ'))

degree(gen=0)

Returns degree of f in x_j.

The degree of 0 is negative infinity.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + y*x + 1, x, y).degree()
2
>>> Poly(x**2 + y*x + y, x, y).degree(y)
1
>>> Poly(0, x).degree()
-oo

degree_list()

Returns a list of degrees of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + y*x + 1, x, y).degree_list()
(2, 1)

diff(*specs, **kwargs)

Computes partial derivative of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 2*x + 1, x).diff()
Poly(2*x + 2, x, domain='ZZ')

>>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))
Poly(2*x*y, x, y, domain='ZZ')

discriminant()

Computes the discriminant of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 2*x + 3, x).discriminant()
-8

dispersion(g=None)

Compute the dispersion of polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion \(\operatorname{dis}(f, g)\) is defined as:

\[\begin{split}\operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \}\end{split}\]

and for a single polynomial \(\operatorname{dis}(f) := \operatorname{dis}(f, f)\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersionset

References

1. [ManWright94]

2. [Koepf98]

3. [Abramov71]

4. [Man93]

dispersionset(g=None)

Compute the dispersion set of two polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion set \(\operatorname{J}(f, g)\) is defined as:

\[\begin{split}\operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}\end{split}\]

For a single polynomial one defines \(\operatorname{J}(f) := \operatorname{J}(f, f)\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersion

References

1. [ManWright94]

2. [Koepf98]

3. [Abramov71]

4. [Man93]

div(g, auto=True)

Polynomial division with remainder of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x))
(Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ'))

>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False)
(Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ'))

domain

Get the ground domain of self.

eject(*gens)

Eject selected generators into the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)

>>> f.eject(x)
Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')
>>> f.eject(y)
Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')

eval(x, a=None, auto=True)

Evaluate f at a in the given variable.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y, z

>>> Poly(x**2 + 2*x + 3, x).eval(2)
11

>>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2)
Poly(5*y + 8, y, domain='ZZ')

>>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)

>>> f.eval({x: 2})
Poly(5*y + 2*z + 6, y, z, domain='ZZ')
>>> f.eval({x: 2, y: 5})
Poly(2*z + 31, z, domain='ZZ')
>>> f.eval({x: 2, y: 5, z: 7})
45

>>> f.eval((2, 5))
Poly(2*z + 31, z, domain='ZZ')
>>> f(2, 5)
Poly(2*z + 31, z, domain='ZZ')

exclude()

Remove unnecessary generators from f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import a, b, c, d, x

>>> Poly(a + x, a, b, c, d, x).exclude()
Poly(a + x, a, x, domain='ZZ')

exquo(g, auto=True)

Computes polynomial exact quotient of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')

>>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1

exquo_ground(coeff)

Exact quotient of f by a an element of the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x + 4).exquo_ground(2)
Poly(x + 2, x, domain='ZZ')

>>> Poly(2*x + 3).exquo_ground(2)
Traceback (most recent call last):
...
ExactQuotientFailed: 2 does not divide 3 in ZZ

factor_list()

Returns a list of irreducible factors of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y

>>> Poly(f).factor_list()
(2, [(Poly(x + y, x, y, domain='ZZ'), 1),
     (Poly(x**2 + 1, x, y, domain='ZZ'), 2)])

factor_list_include()

Returns a list of irreducible factors of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y

>>> Poly(f).factor_list_include()
[(Poly(2*x + 2*y, x, y, domain='ZZ'), 1),
 (Poly(x**2 + 1, x, y, domain='ZZ'), 2)]

free_symbols

Free symbols of a polynomial expression.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y, z

>>> Poly(x**2 + 1).free_symbols
{x}
>>> Poly(x**2 + y).free_symbols
{x, y}
>>> Poly(x**2 + y, x).free_symbols
{x, y}
>>> Poly(x**2 + y, x, z).free_symbols
{x, y}

free_symbols_in_domain

Free symbols of the domain of self.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 1).free_symbols_in_domain
set()
>>> Poly(x**2 + y).free_symbols_in_domain
set()
>>> Poly(x**2 + y, x).free_symbols_in_domain
{y}

classmethod from_dict(rep, *gens, **args)

Construct a polynomial from a dict.

classmethod from_expr(rep, *gens, **args)

Construct a polynomial from an expression.

classmethod from_list(rep, *gens, **args)

Construct a polynomial from a list.

classmethod from_poly(rep, *gens, **args)

Construct a polynomial from a polynomial.

gcd(g)

Returns the polynomial GCD of f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x))
Poly(x - 1, x, domain='ZZ')

gcdex(g, auto=True)

Extended Euclidean algorithm of f and g.

Returns (s, t, h) such that h = gcd(f, g) and s*f + t*g = h.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4

>>> Poly(f).gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'),
 Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'),
 Poly(x + 1, x, domain='QQ'))

gen

Return the principal generator.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).gen
x

get_domain()

Get the ground domain of f.

get_modulus()

Get the modulus of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, modulus=2).get_modulus()
2

gff_list()

Computes greatest factorial factorization of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = x**5 + 2*x**4 - x**3 - 2*x**2

>>> Poly(f).gff_list()
[(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]

ground_roots()

Compute roots of f by factorization in the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots()
{0: 2, 1: 2}

half_gcdex(g, auto=True)

Half extended Euclidean algorithm of f and g.

Returns (s, h) such that h = gcd(f, g) and s*f = h (mod g).

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4

>>> Poly(f).half_gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ'))

has_only_gens(*gens)

Return True if Poly(f, *gens) retains ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y, z

>>> Poly(x*y + 1, x, y, z).has_only_gens(x, y)
True
>>> Poly(x*y + z, x, y, z).has_only_gens(x, y)
False

homogeneous_order()

Returns the homogeneous order of f.

A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. This degree is the homogeneous order of f. If you only want to check if a polynomial is homogeneous, then use Poly.is_homogeneous().

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4)
>>> f.homogeneous_order()
5

homogenize(s)

Returns the homogeneous polynomial of f.

A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. If you only want to check if a polynomial is homogeneous, then use Poly.is_homogeneous(). If you want not only to check if a polynomial is homogeneous but also compute its homogeneous order, then use Poly.homogeneous_order().

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y, z

>>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3)
>>> f.homogenize(z)
Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ')

inject(front=False)

Inject ground domain generators into f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x)

>>> f.inject()
Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ')
>>> f.inject(front=True)
Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ')

integrate(*specs, **args)

Computes indefinite integral of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 2*x + 1, x).integrate()
Poly(1/3*x**3 + x**2 + x, x, domain='QQ')

>>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0))
Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ')

intervals(all=False, eps=None, inf=None, sup=None, fast=False, sqf=False)

Compute isolating intervals for roots of f.

For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 3, x).intervals()
[((-2, -1), 1), ((1, 2), 1)]
>>> Poly(x**2 - 3, x).intervals(eps=1e-2)
[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]

References

1

Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.

2

Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.

invert(g, auto=True)

Invert f modulo g when possible.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x))
Poly(-4/3, x, domain='QQ')

>>> Poly(x**2 - 1, x).invert(Poly(x - 1, x))
Traceback (most recent call last):
...
NotInvertible: zero divisor

is_cyclotomic

Returns True if f is a cyclotomic polnomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1

>>> Poly(f).is_cyclotomic
False

>>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1

>>> Poly(g).is_cyclotomic
True

is_ground

Returns True if f is an element of the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x, x).is_ground
False
>>> Poly(2, x).is_ground
True
>>> Poly(y, x).is_ground
True

is_homogeneous

Returns True if f is a homogeneous polynomial.

A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. If you want not only to check if a polynomial is homogeneous but also compute its homogeneous order, then use Poly.homogeneous_order().

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + x*y, x, y).is_homogeneous
True
>>> Poly(x**3 + x*y, x, y).is_homogeneous
False

is_irreducible

Returns True if f has no factors over its domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible
True
>>> Poly(x**2 + 1, x, modulus=2).is_irreducible
False

is_linear

Returns True if f is linear in all its variables.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x + y + 2, x, y).is_linear
True
>>> Poly(x*y + 2, x, y).is_linear
False

is_monic

Returns True if the leading coefficient of f is one.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x + 2, x).is_monic
True
>>> Poly(2*x + 2, x).is_monic
False

is_monomial

Returns True if f is zero or has only one term.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(3*x**2, x).is_monomial
True
>>> Poly(3*x**2 + 1, x).is_monomial
False

is_multivariate

Returns True if f is a multivariate polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + x + 1, x).is_multivariate
False
>>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate
True
>>> Poly(x*y**2 + x*y + 1, x).is_multivariate
False
>>> Poly(x**2 + x + 1, x, y).is_multivariate
True

is_one

Returns True if f is a unit polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(0, x).is_one
False
>>> Poly(1, x).is_one
True

is_primitive

Returns True if GCD of the coefficients of f is one.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**2 + 6*x + 12, x).is_primitive
False
>>> Poly(x**2 + 3*x + 6, x).is_primitive
True

is_quadratic

Returns True if f is quadratic in all its variables.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x*y + 2, x, y).is_quadratic
True
>>> Poly(x*y**2 + 2, x, y).is_quadratic
False

is_sqf

Returns True if f is a square-free polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 2*x + 1, x).is_sqf
False
>>> Poly(x**2 - 1, x).is_sqf
True

is_univariate

Returns True if f is a univariate polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + x + 1, x).is_univariate
True
>>> Poly(x*y**2 + x*y + 1, x, y).is_univariate
False
>>> Poly(x*y**2 + x*y + 1, x).is_univariate
True
>>> Poly(x**2 + x + 1, x, y).is_univariate
False

is_zero

Returns True if f is a zero polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(0, x).is_zero
True
>>> Poly(1, x).is_zero
False

l1_norm()

Returns l1 norm of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(-x**2 + 2*x - 3, x).l1_norm()
6

lcm(g)

Returns polynomial LCM of f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x))
Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ')

length()

Returns the number of non-zero terms in f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 2*x - 1).length()
3

lift()

Convert algebraic coefficients to rationals.

Examples

>>> from sympy import Poly, I
>>> from sympy.abc import x

>>> Poly(x**2 + I*x + 1, x, extension=I).lift()
Poly(x**4 + 3*x**2 + 1, x, domain='QQ')

ltrim(gen)

Remove dummy generators from f that are to the left of specified gen in the generators as ordered. When gen is an integer, it refers to the generator located at that position within the tuple of generators of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y, z

>>> Poly(y**2 + y*z**2, x, y, z).ltrim(y)
Poly(y**2 + y*z**2, y, z, domain='ZZ')
>>> Poly(z, x, y, z).ltrim(-1)
Poly(z, z, domain='ZZ')

max_norm()

Returns maximum norm of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(-x**2 + 2*x - 3, x).max_norm()
3

monic(auto=True)

Divides all coefficients by LC(f).

Examples

>>> from sympy import Poly, ZZ
>>> from sympy.abc import x

>>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic()
Poly(x**2 + 2*x + 3, x, domain='QQ')

>>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic()
Poly(x**2 + 4/3*x + 2/3, x, domain='QQ')

monoms(order=None)

Returns all non-zero monomials from f in lex order.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms()
[(2, 0), (1, 2), (1, 1), (0, 1)]

See also

all_monoms

mul(g)

Multiply two polynomials f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).mul(Poly(x - 2, x))
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')

>>> Poly(x**2 + 1, x)*Poly(x - 2, x)
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')

mul_ground(coeff)

Multiply f by a an element of the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x + 1).mul_ground(2)
Poly(2*x + 2, x, domain='ZZ')

neg()

Negate all coefficients in f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).neg()
Poly(-x**2 + 1, x, domain='ZZ')

>>> -Poly(x**2 - 1, x)
Poly(-x**2 + 1, x, domain='ZZ')

classmethod new(rep, *gens)

Construct Poly instance from raw representation.

norm()

Computes the product, Norm(f), of the conjugates of a polynomial f defined over a number field K.

Examples

>>> from sympy import Poly, sqrt
>>> from sympy.abc import x

>>> a, b = sqrt(2), sqrt(3)

A polynomial over a quadratic extension. Two conjugates x - a and x + a.

>>> f = Poly(x - a, x, extension=a)
>>> f.norm()
Poly(x**2 - 2, x, domain='QQ')

A polynomial over a quartic extension. Four conjugates x - a, x - a, x + a and x + a.

>>> f = Poly(x - a, x, extension=(a, b))
>>> f.norm()
Poly(x**4 - 4*x**2 + 4, x, domain='QQ')

nroots(n=15, maxsteps=50, cleanup=True)

Compute numerical approximations of roots of f.

Parameters

n … the number of digits to calculate

maxsteps … the maximum number of iterations to do

If the accuracy `n` cannot be reached in `maxsteps`, it will raise an

exception. You need to rerun with higher maxsteps.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 3).nroots(n=15)
[-1.73205080756888, 1.73205080756888]
>>> Poly(x**2 - 3).nroots(n=30)
[-1.73205080756887729352744634151, 1.73205080756887729352744634151]

nth(*N)

Returns the n-th coefficient of f where N are the exponents of the generators in the term of interest.

Examples

>>> from sympy import Poly, sqrt
>>> from sympy.abc import x, y

>>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2)
2
>>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2)
2
>>> Poly(4*sqrt(x)*y)
Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ')
>>> _.nth(1, 1)
4

See also

coeff_monomial

nth_power_roots_poly(n)

Construct a polynomial with n-th powers of roots of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = Poly(x**4 - x**2 + 1)

>>> f.nth_power_roots_poly(2)
Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(3)
Poly(x**4 + 2*x**2 + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(4)
Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(12)
Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ')

one

Return one polynomial with self’s properties.

pdiv(g)

Polynomial pseudo-division of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x))
(Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ'))

per(rep, gens=None, remove=None)

Create a Poly out of the given representation.

Examples

>>> from sympy import Poly, ZZ
>>> from sympy.abc import x, y

>>> from sympy.polys.polyclasses import DMP

>>> a = Poly(x**2 + 1)

>>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y])
Poly(y + 1, y, domain='ZZ')

pexquo(g)

Polynomial exact pseudo-quotient of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')

>>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1

pow(n)

Raise f to a non-negative power n.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x - 2, x).pow(3)
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')

>>> Poly(x - 2, x)**3
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')

pquo(g)

Polynomial pseudo-quotient of f by g.

See the Caveat note in the function prem(f, g).

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x))
Poly(2*x + 4, x, domain='ZZ')

>>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')

prem(g)

Polynomial pseudo-remainder of f by g.

Caveat: The function prem(f, g, x) can be safely used to compute

in Z[x] _only_ subresultant polynomial remainder sequences (prs’s).

To safely compute Euclidean and Sturmian prs’s in Z[x] employ anyone of the corresponding functions found in the module sympy.polys.subresultants_qq_zz. The functions in the module with suffix _pg compute prs’s in Z[x] employing rem(f, g, x), whereas the functions with suffix _amv compute prs’s in Z[x] employing rem_z(f, g, x).

The function rem_z(f, g, x) differs from prem(f, g, x) in that to compute the remainder polynomials in Z[x] it premultiplies the divident times the absolute value of the leading coefficient of the divisor raised to the power degree(f, x) - degree(g, x) + 1.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x))
Poly(20, x, domain='ZZ')

primitive()

Returns the content and a primitive form of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**2 + 8*x + 12, x).primitive()
(2, Poly(x**2 + 4*x + 6, x, domain='ZZ'))

quo(g, auto=True)

Computes polynomial quotient of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x))
Poly(1/2*x + 1, x, domain='QQ')

>>> Poly(x**2 - 1, x).quo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')

quo_ground(coeff)

Quotient of f by a an element of the ground domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x + 4).quo_ground(2)
Poly(x + 2, x, domain='ZZ')

>>> Poly(2*x + 3).quo_ground(2)
Poly(x + 1, x, domain='ZZ')

rat_clear_denoms(g)

Clear denominators in a rational function f/g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> f = Poly(x**2/y + 1, x)
>>> g = Poly(x**3 + y, x)

>>> p, q = f.rat_clear_denoms(g)

>>> p
Poly(x**2 + y, x, domain='ZZ[y]')
>>> q
Poly(y*x**3 + y**2, x, domain='ZZ[y]')

real_roots(multiple=True, radicals=True)

Return a list of real roots with multiplicities.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).real_roots()
[CRootOf(x**3 + x + 1, 0)]

refine_root(s, t, eps=None, steps=None, fast=False, check_sqf=False)

Refine an isolating interval of a root to the given precision.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2)
(19/11, 26/15)

rem(g, auto=True)

Computes the polynomial remainder of f by g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x))
Poly(5, x, domain='ZZ')

>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False)
Poly(x**2 + 1, x, domain='ZZ')

reorder(*gens, **args)

Efficiently apply new order of generators.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + x*y**2, x, y).reorder(y, x)
Poly(y**2*x + x**2, y, x, domain='ZZ')

replace(x, y=None, *_ignore)

Replace x with y in generators list.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 1, x).replace(x, y)
Poly(y**2 + 1, y, domain='ZZ')

resultant(g, includePRS=False)

Computes the resultant of f and g via PRS.

If includePRS=True, it includes the subresultant PRS in the result. Because the PRS is used to calculate the resultant, this is more efficient than calling subresultants() separately.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = Poly(x**2 + 1, x)

>>> f.resultant(Poly(x**2 - 1, x))
4
>>> f.resultant(Poly(x**2 - 1, x), includePRS=True)
(4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'),
     Poly(-2, x, domain='ZZ')])

retract(field=None)

Recalculate the ground domain of a polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = Poly(x**2 + 1, x, domain='QQ[y]')
>>> f
Poly(x**2 + 1, x, domain='QQ[y]')

>>> f.retract()
Poly(x**2 + 1, x, domain='ZZ')
>>> f.retract(field=True)
Poly(x**2 + 1, x, domain='QQ')

revert(n)

Compute f**(-1) mod x**n.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(1, x).revert(2)
Poly(1, x, domain='ZZ')

>>> Poly(1 + x, x).revert(1)
Poly(1, x, domain='ZZ')

>>> Poly(x**2 - 1, x).revert(1)
Traceback (most recent call last):
...
NotReversible: only unity is reversible in a ring

>>> Poly(1/x, x).revert(1)
Traceback (most recent call last):
...
PolynomialError: 1/x contains an element of the generators set

root(index, radicals=True)

Get an indexed root of a polynomial.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4)

>>> f.root(0)
-1/2
>>> f.root(1)
2
>>> f.root(2)
2
>>> f.root(3)
Traceback (most recent call last):
...
IndexError: root index out of [-3, 2] range, got 3

>>> Poly(x**5 + x + 1).root(0)
CRootOf(x**3 - x**2 + 1, 0)

set_domain(domain)

Set the ground domain of f.

set_modulus(modulus)

Set the modulus of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2)
Poly(x**2 + 1, x, modulus=2)

shift(a)

Efficiently compute Taylor shift f(x + a).

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 2*x + 1, x).shift(2)
Poly(x**2 + 2*x + 1, x, domain='ZZ')

slice(x, m, n=None)

Take a continuous subsequence of terms of f.

sqf_list(all=False)

Returns a list of square-free factors of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16

>>> Poly(f).sqf_list()
(2, [(Poly(x + 1, x, domain='ZZ'), 2),
     (Poly(x + 2, x, domain='ZZ'), 3)])

>>> Poly(f).sqf_list(all=True)
(2, [(Poly(1, x, domain='ZZ'), 1),
     (Poly(x + 1, x, domain='ZZ'), 2),
     (Poly(x + 2, x, domain='ZZ'), 3)])

sqf_list_include(all=False)

Returns a list of square-free factors of f.

Examples

>>> from sympy import Poly, expand
>>> from sympy.abc import x

>>> f = expand(2*(x + 1)**3*x**4)
>>> f
2*x**7 + 6*x**6 + 6*x**5 + 2*x**4

>>> Poly(f).sqf_list_include()
[(Poly(2, x, domain='ZZ'), 1),
 (Poly(x + 1, x, domain='ZZ'), 3),
 (Poly(x, x, domain='ZZ'), 4)]

>>> Poly(f).sqf_list_include(all=True)
[(Poly(2, x, domain='ZZ'), 1),
 (Poly(1, x, domain='ZZ'), 2),
 (Poly(x + 1, x, domain='ZZ'), 3),
 (Poly(x, x, domain='ZZ'), 4)]

sqf_norm()

Computes square-free norm of f.

Returns s, f, r, such that g(x) = f(x-sa) and r(x) = Norm(g(x)) is a square-free polynomial over K, where a is the algebraic extension of the ground domain.

Examples

>>> from sympy import Poly, sqrt
>>> from sympy.abc import x

>>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm()

>>> s
1
>>> f
Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ<sqrt(3)>')
>>> r
Poly(x**4 - 4*x**2 + 16, x, domain='QQ')

sqf_part()

Computes square-free part of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 - 3*x - 2, x).sqf_part()
Poly(x**2 - x - 2, x, domain='ZZ')

sqr()

Square a polynomial f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x - 2, x).sqr()
Poly(x**2 - 4*x + 4, x, domain='ZZ')

>>> Poly(x - 2, x)**2
Poly(x**2 - 4*x + 4, x, domain='ZZ')

sturm(auto=True)

Computes the Sturm sequence of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**3 - 2*x**2 + x - 3, x).sturm()
[Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'),
 Poly(3*x**2 - 4*x + 1, x, domain='QQ'),
 Poly(2/9*x + 25/9, x, domain='QQ'),
 Poly(-2079/4, x, domain='QQ')]

sub(g)

Subtract two polynomials f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).sub(Poly(x - 2, x))
Poly(x**2 - x + 3, x, domain='ZZ')

>>> Poly(x**2 + 1, x) - Poly(x - 2, x)
Poly(x**2 - x + 3, x, domain='ZZ')

sub_ground(coeff)

Subtract an element of the ground domain from f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x + 1).sub_ground(2)
Poly(x - 1, x, domain='ZZ')

subresultants(g)

Computes the subresultant PRS of f and g.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x))
[Poly(x**2 + 1, x, domain='ZZ'),
 Poly(x**2 - 1, x, domain='ZZ'),
 Poly(-2, x, domain='ZZ')]

terms(order=None)

Returns all non-zero terms from f in lex order.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms()
[((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]

See also

all_terms

terms_gcd()

Remove GCD of terms from the polynomial f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd()
((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ'))

termwise(func, *gens, **args)

Apply a function to all terms of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> def func(k, coeff):
...     k = k[0]
...     return coeff//10**(2-k)

>>> Poly(x**2 + 20*x + 400).termwise(func)
Poly(x**2 + 2*x + 4, x, domain='ZZ')

to_exact()

Make the ground domain exact.

Examples

>>> from sympy import Poly, RR
>>> from sympy.abc import x

>>> Poly(x**2 + 1.0, x, domain=RR).to_exact()
Poly(x**2 + 1, x, domain='QQ')

to_field()

Make the ground domain a field.

Examples

>>> from sympy import Poly, ZZ
>>> from sympy.abc import x

>>> Poly(x**2 + 1, x, domain=ZZ).to_field()
Poly(x**2 + 1, x, domain='QQ')

to_ring()

Make the ground domain a ring.

Examples

>>> from sympy import Poly, QQ
>>> from sympy.abc import x

>>> Poly(x**2 + 1, domain=QQ).to_ring()
Poly(x**2 + 1, x, domain='ZZ')

total_degree()

Returns the total degree of f.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x, y

>>> Poly(x**2 + y*x + 1, x, y).total_degree()
2
>>> Poly(x + y**5, x, y).total_degree()
5

transform(p, q)

Efficiently evaluate the functional transformation q**n * f(p/q).

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x))
Poly(4, x, domain='ZZ')

trunc(p)

Reduce f modulo a constant p.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3)
Poly(-x**3 - x + 1, x, domain='ZZ')

unify(g)

Make f and g belong to the same domain.

Examples

>>> from sympy import Poly
>>> from sympy.abc import x

>>> f, g = Poly(x/2 + 1), Poly(2*x + 1)

>>> f
Poly(1/2*x + 1, x, domain='QQ')
>>> g
Poly(2*x + 1, x, domain='ZZ')

>>> F, G = f.unify(g)

>>> F
Poly(1/2*x + 1, x, domain='QQ')
>>> G
Poly(2*x + 1, x, domain='QQ')

unit

Return unit polynomial with self’s properties.

zero

Return zero polynomial with self’s properties.

class sympy.polys.polytools.PurePoly

Class for representing pure polynomials.

free_symbols

Free symbols of a polynomial.

Examples

>>> from sympy import PurePoly
>>> from sympy.abc import x, y

>>> PurePoly(x**2 + 1).free_symbols
set()
>>> PurePoly(x**2 + y).free_symbols
set()
>>> PurePoly(x**2 + y, x).free_symbols
{y}

class sympy.polys.polytools.GroebnerBasis

Represents a reduced Groebner basis.

contains(poly)

Check if poly belongs the ideal generated by self.

Examples

>>> from sympy import groebner
>>> from sympy.abc import x, y

>>> f = 2*x**3 + y**3 + 3*y
>>> G = groebner([x**2 + y**2 - 1, x*y - 2])

>>> G.contains(f)
True
>>> G.contains(f + 1)
False

fglm(order)

Convert a Groebner basis from one ordering to another.

The FGLM algorithm converts reduced Groebner bases of zero-dimensional ideals from one ordering to another. This method is often used when it is infeasible to compute a Groebner basis with respect to a particular ordering directly.

Examples

>>> from sympy.abc import x, y
>>> from sympy import groebner

>>> F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
>>> G = groebner(F, x, y, order='grlex')

>>> list(G.fglm('lex'))
[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]
>>> list(groebner(F, x, y, order='lex'))
[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]

References

R557

J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient Computation of Zero-dimensional Groebner Bases by Change of Ordering

is_zero_dimensional

Checks if the ideal generated by a Groebner basis is zero-dimensional.

The algorithm checks if the set of monomials not divisible by the leading monomial of any element of F is bounded.

References

David A. Cox, John B. Little, Donal O’Shea. Ideals, Varieties and Algorithms, 3rd edition, p. 230

reduce(expr, auto=True)

Reduces a polynomial modulo a Groebner basis.

Given a polynomial f and a set of polynomials G = (g_1, ..., g_n), computes a set of quotients q = (q_1, ..., q_n) and the remainder r such that f = q_1*f_1 + ... + q_n*f_n + r, where r vanishes or r is a completely reduced polynomial with respect to G.

Examples

>>> from sympy import groebner, expand
>>> from sympy.abc import x, y

>>> f = 2*x**4 - x**2 + y**3 + y**2
>>> G = groebner([x**3 - x, y**3 - y])

>>> G.reduce(f)
([2*x, 1], x**2 + y**2 + y)
>>> Q, r = _

>>> expand(sum(q*g for q, g in zip(Q, G)) + r)
2*x**4 - x**2 + y**3 + y**2
>>> _ == f
True

Extra polynomial manipulation functions

sympy.polys.polyfuncs.symmetrize(F, *gens, **args)

Rewrite a polynomial in terms of elementary symmetric polynomials.

A symmetric polynomial is a multivariate polynomial that remains invariant under any variable permutation, i.e., if f = f(x_1, x_2, ..., x_n), then f = f(x_{i_1}, x_{i_2}, ..., x_{i_n}), where (i_1, i_2, ..., i_n) is a permutation of (1, 2, ..., n) (an element of the group S_n).

Returns a tuple of symmetric polynomials (f1, f2, ..., fn) such that f = f1 + f2 + ... + fn.

Examples

>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y

>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)

>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])

>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)

>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])

sympy.polys.polyfuncs.horner(f, *gens, **args)

Rewrite a polynomial in Horner form.

Among other applications, evaluation of a polynomial at a point is optimal when it is applied using the Horner scheme ([1]).

Examples

>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e

>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5

>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))

>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y

>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))

>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))

References

[1] - https://en.wikipedia.org/wiki/Horner_scheme

sympy.polys.polyfuncs.interpolate(data, x)

Construct an interpolating polynomial for the data points.

Examples

>>> from sympy.polys.polyfuncs import interpolate
>>> from sympy.abc import x

A list is interpreted as though it were paired with a range starting from 1:

>>> interpolate([1, 4, 9, 16], x)
x**2

This can be made explicit by giving a list of coordinates:

>>> interpolate([(1, 1), (2, 4), (3, 9)], x)
x**2

The (x, y) coordinates can also be given as keys and values of a dictionary (and the points need not be equispaced):

>>> interpolate([(-1, 2), (1, 2), (2, 5)], x)
x**2 + 1
>>> interpolate({-1: 2, 1: 2, 2: 5}, x)
x**2 + 1

sympy.polys.polyfuncs.viete(f, roots=None, *gens, **args)

Generate Viete’s formulas for f.

Examples

>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols

>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')

>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]

Domain constructors

sympy.polys.constructor.construct_domain(obj, **args)

Construct a minimal domain for the list of coefficients.

Algebraic number fields

sympy.polys.numberfields.minimal_polynomial(ex, x=None, compose=True, polys=False, domain=None)

Computes the minimal polynomial of an algebraic element.

Parameters

ex : Expr

Element or expression whose minimal polynomial is to be calculated.

x : Symbol, optional

Independent variable of the minimal polynomial

compose : boolean, optional (default=True)

Method to use for computing minimal polynomial. If compose=True (default) then _minpoly_compose is used, if compose=False then groebner bases are used.

polys : boolean, optional (default=False)

If True returns a Poly object else an Expr object.

domain : Domain, optional

Ground domain

Notes

By default compose=True, the minimal polynomial of the subexpressions of ex are computed, then the arithmetic operations on them are performed using the resultant and factorization. If compose=False, a bottom-up algorithm is used with groebner. The default algorithm stalls less frequently.

If no ground domain is given, it will be generated automatically from the expression.

Examples

>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y

>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y

sympy.polys.numberfields.minpoly(ex, x=None, compose=True, polys=False, domain=None)

Computes the minimal polynomial of an algebraic element.

Parameters

ex : Expr

Element or expression whose minimal polynomial is to be calculated.

x : Symbol, optional

Independent variable of the minimal polynomial

compose : boolean, optional (default=True)

Method to use for computing minimal polynomial. If compose=True (default) then _minpoly_compose is used, if compose=False then groebner bases are used.

polys : boolean, optional (default=False)

If True returns a Poly object else an Expr object.

domain : Domain, optional

Ground domain

Notes

By default compose=True, the minimal polynomial of the subexpressions of ex are computed, then the arithmetic operations on them are performed using the resultant and factorization. If compose=False, a bottom-up algorithm is used with groebner. The default algorithm stalls less frequently.

If no ground domain is given, it will be generated automatically from the expression.

Examples

>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y

>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y

sympy.polys.numberfields.primitive_element(extension, x=None, **args)

Construct a common number field for all extensions.

sympy.polys.numberfields.field_isomorphism(a, b, **args)

Construct an isomorphism between two number fields.

sympy.polys.numberfields.to_number_field(extension, theta=None, **args)

Express \(extension\) in the field generated by \(theta\).

sympy.polys.numberfields.isolate(alg, eps=None, fast=False)

Give a rational isolating interval for an algebraic number.

class sympy.polys.numberfields.AlgebraicNumber

Class for representing algebraic numbers in SymPy.

as_expr(x=None)

Create a Basic expression from self.

as_poly(x=None)

Create a Poly instance from self.

coeffs()

Returns all SymPy coefficients of an algebraic number.

is_aliased

Returns True if alias was set.

native_coeffs()

Returns all native coefficients of an algebraic number.

to_algebraic_integer()

Convert self to an algebraic integer.

Monomials encoded as tuples

class sympy.polys.monomials.Monomial(monom, gens=None)

Class representing a monomial, i.e. a product of powers.

sympy.polys.monomials.itermonomials(variables, max_degree, min_degree=0)

Generate a set of monomials of the degree greater than or equal to \(min_degree\) and less than or equal to \(max_degree\).

Given a set of variables \(V\) and a min_degree \(N\) and a max_degree \(M\) generate a set of monomials of degree less than or equal to \(N\) and greater than or equal to \(M\). The total number of monomials in commutative variables is huge and is given by the following formula if \(M = 0\):

\[\frac{(\#V + N)!}{\#V! N!}\]

For example if we would like to generate a dense polynomial of a total degree \(N = 50\) and \(M = 0\), which is the worst case, in 5 variables, assuming that exponents and all of coefficients are 32-bit long and stored in an array we would need almost 80 GiB of memory! Fortunately most polynomials, that we will encounter, are sparse.

Examples

Consider monomials in commutative variables \(x\) and \(y\) and non-commutative variables \(a\) and \(b\):

>>> from sympy import symbols
>>> from sympy.polys.monomials import itermonomials
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y

>>> sorted(itermonomials([x, y], 2), key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2]

>>> sorted(itermonomials([x, y], 3), key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3]

>>> a, b = symbols('a, b', commutative=False)
>>> itermonomials([a, b, x], 2)
{1, a, a**2, b, b**2, x, x**2, a*b, b*a, x*a, x*b}

>>> sorted(itermonomials([x, y], 2, 1), key=monomial_key('grlex', [y, x]))
[x, y, x**2, x*y, y**2]

sympy.polys.monomials.monomial_count(V, N)

Computes the number of monomials.

The number of monomials is given by the following formula:

\[\frac{(\#V + N)!}{\#V! N!}\]

where \(N\) is a total degree and \(V\) is a set of variables.

Examples

>>> from sympy.polys.monomials import itermonomials, monomial_count
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y

>>> monomial_count(2, 2)
6

>>> M = itermonomials([x, y], 2)

>>> sorted(M, key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2]
>>> len(M)
6

Orderings of monomials

class sympy.polys.orderings.LexOrder

Lexicographic order of monomials.

class sympy.polys.orderings.GradedLexOrder

Graded lexicographic order of monomials.

class sympy.polys.orderings.ReversedGradedLexOrder

Reversed graded lexicographic order of monomials.

Formal manipulation of roots of polynomials

sympy.polys.rootoftools.rootof(f, x, index=None, radicals=True, expand=True)

An indexed root of a univariate polynomial.

Returns either a ComplexRootOf object or an explicit expression involving radicals.

Parameters

f : Expr

Univariate polynomial.

x : Symbol, optional

Generator for f.

index : int or Integer

radicals : bool

Return a radical expression if possible.

expand : bool

Expand f.

class sympy.polys.rootoftools.RootOf

Represents a root of a univariate polynomial.

Base class for roots of different kinds of polynomials. Only complex roots are currently supported.

class sympy.polys.rootoftools.ComplexRootOf

Represents an indexed complex root of a polynomial.

Roots of a univariate polynomial separated into disjoint real or complex intervals and indexed in a fixed order. Currently only rational coefficients are allowed. Can be imported as CRootOf. To avoid confusion, the generator must be a Symbol.

Examples

>>> from sympy import CRootOf, rootof
>>> from sympy.abc import x

CRootOf is a way to reference a particular root of a polynomial. If there is a rational root, it will be returned:

>>> CRootOf.clear_cache()  # for doctest reproducibility
>>> CRootOf(x**2 - 4, 0)
-2

Whether roots involving radicals are returned or not depends on whether the radicals flag is true (which is set to True with rootof):

>>> CRootOf(x**2 - 3, 0)
CRootOf(x**2 - 3, 0)
>>> CRootOf(x**2 - 3, 0, radicals=True)
-sqrt(3)
>>> rootof(x**2 - 3, 0)
-sqrt(3)

The following cannot be expressed in terms of radicals:

>>> r = rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0); r
CRootOf(4*x**5 + 16*x**3 + 12*x**2 + 7, 0)

The root bounds can be seen, however, and they are used by the evaluation methods to get numerical approximations for the root.

>>> interval = r._get_interval(); interval
(-1, 0)
>>> r.evalf(2)
-0.98

The evalf method refines the width of the root bounds until it guarantees that any decimal approximation within those bounds will satisfy the desired precision. It then stores the refined interval so subsequent requests at or below the requested precision will not have to recompute the root bounds and will return very quickly.

Before evaluation above, the interval was

>>> interval
(-1, 0)

After evaluation it is now

>>. r._get_interval() (-165/169, -206/211)

To reset all intervals for a given polynomial, the \(_reset\) method can be called from any CRootOf instance of the polynomial:

>>> r._reset()
>>> r._get_interval()
(-1, 0)

The \(eval_approx\) method will also find the root to a given precision but the interval is not modified unless the search for the root fails to converge within the root bounds. And the secant method is used to find the root. (The evalf method uses bisection and will always update the interval.)

>>> r.eval_approx(2)
-0.98

The interval needed to be slightly updated to find that root:

>>> r._get_interval()
(-1, -1/2)

The evalf_rational will compute a rational approximation of the root to the desired accuracy or precision.

>>> r.eval_rational(n=2)
-69629/71318

>>> t = CRootOf(x**3 + 10*x + 1, 1)
>>> t.eval_rational(1e-1)
15/256 - 805*I/256
>>> t.eval_rational(1e-1, 1e-4)
3275/65536 - 414645*I/131072
>>> t.eval_rational(1e-4, 1e-4)
6545/131072 - 414645*I/131072
>>> t.eval_rational(n=2)
104755/2097152 - 6634255*I/2097152

Notes

Although a PurePoly can be constructed from a non-symbol generator RootOf instances of non-symbols are disallowed to avoid confusion over what root is being represented.

>>> from sympy import exp, PurePoly
>>> PurePoly(x) == PurePoly(exp(x))
True
>>> CRootOf(x - 1, 0)
1
>>> CRootOf(exp(x) - 1, 0)  # would correspond to x == 0
Traceback (most recent call last):
...
sympy.polys.polyerrors.PolynomialError: generator must be a Symbol

See also

eval_approx, eval_rational, _eval_evalf

class sympy.polys.rootoftools.RootSum

Represents a sum of all roots of a univariate polynomial.

Symbolic root-finding algorithms

sympy.polys.polyroots.roots(f, *gens, **flags)

Computes symbolic roots of a univariate polynomial.

Given a univariate polynomial f with symbolic coefficients (or a list of the polynomial’s coefficients), returns a dictionary with its roots and their multiplicities.

Only roots expressible via radicals will be returned. To get a complete set of roots use RootOf class or numerical methods instead. By default cubic and quartic formulas are used in the algorithm. To disable them because of unreadable output set cubics=False or quartics=False respectively. If cubic roots are real but are expressed in terms of complex numbers (casus irreducibilis [1]) the trig flag can be set to True to have the solutions returned in terms of cosine and inverse cosine functions.

To get roots from a specific domain set the filter flag with one of the following specifiers: Z, Q, R, I, C. By default all roots are returned (this is equivalent to setting filter='C').

By default a dictionary is returned giving a compact result in case of multiple roots. However to get a list containing all those roots set the multiple flag to True; the list will have identical roots appearing next to each other in the result. (For a given Poly, the all_roots method will give the roots in sorted numerical order.)

Examples

>>> from sympy import Poly, roots
>>> from sympy.abc import x, y

>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}

>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}

>>> p = Poly(x**2-y, x, y)

>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}

>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}

>>> roots([1, 0, -1])
{-1: 1, 1: 1}

References

R558

https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method

Special polynomials

sympy.polys.specialpolys.swinnerton_dyer_poly(n, x=None, polys=False)

Generates n-th Swinnerton-Dyer polynomial in \(x\).

Parameters

n : int

\(n\) decides the order of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.specialpolys.interpolating_poly(n, x, X='x', Y='y')

Construct Lagrange interpolating polynomial for n data points.

sympy.polys.specialpolys.cyclotomic_poly(n, x=None, polys=False)

Generates cyclotomic polynomial of order \(n\) in \(x\).

Parameters

n : int

\(n\) decides the order of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.specialpolys.symmetric_poly(n, *gens, **args)

Generates symmetric polynomial of order \(n\).

Returns a Poly object when polys=True, otherwise (default) returns an expression.

sympy.polys.specialpolys.random_poly(x, n, inf, sup, domain=ZZ, polys=False)

Generates a polynomial of degree n with coefficients in [inf, sup].

Parameters

x

\(x\) is the independent term of polynomial

n : int

\(n\) decides the order of polynomial

inf

Lower limit of range in which coefficients lie

sup

Upper limit of range in which coefficients lie

domain : optional

Decides what ring the coefficients are supposed to belong. Default is set to Integers.

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

Orthogonal polynomials

sympy.polys.orthopolys.chebyshevt_poly(n, x=None, polys=False)

Generates Chebyshev polynomial of the first kind of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.chebyshevu_poly(n, x=None, polys=False)

Generates Chebyshev polynomial of the second kind of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.gegenbauer_poly(n, a, x=None, polys=False)

Generates Gegenbauer polynomial of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

a

Decides minimal domain for the list of coefficients.

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.hermite_poly(n, x=None, polys=False)

Generates Hermite polynomial of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.jacobi_poly(n, a, b, x=None, polys=False)

Generates Jacobi polynomial of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

a

Lower limit of minimal domain for the list of coefficients.

b

Upper limit of minimal domain for the list of coefficients.

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.legendre_poly(n, x=None, polys=False)

Generates Legendre polynomial of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

sympy.polys.orthopolys.laguerre_poly(n, x=None, alpha=None, polys=False)

Generates Laguerre polynomial of degree \(n\) in \(x\).

Parameters

n : int

\(n\) decides the degree of polynomial

x : optional

alpha

Decides minimal domain for the list of coefficients.

polys : bool, optional

polys=True returns an expression, otherwise (default) returns an expression.

Manipulation of rational functions

sympy.polys.rationaltools.together(expr, deep=False, fraction=True)

Denest and combine rational expressions using symbolic methods.

This function takes an expression or a container of expressions and puts it (them) together by denesting and combining rational subexpressions. No heroic measures are taken to minimize degree of the resulting numerator and denominator. To obtain completely reduced expression use cancel(). However, together() can preserve as much as possible of the structure of the input expression in the output (no expansion is performed).

A wide variety of objects can be put together including lists, tuples, sets, relational objects, integrals and others. It is also possible to transform interior of function applications, by setting deep flag to True.

By definition, together() is a complement to apart(), so apart(together(expr)) should return expr unchanged. Note however, that together() uses only symbolic methods, so it might be necessary to use cancel() to perform algebraic simplification and minimize degree of the numerator and denominator.

Examples

>>> from sympy import together, exp
>>> from sympy.abc import x, y, z

>>> together(1/x + 1/y)
(x + y)/(x*y)
>>> together(1/x + 1/y + 1/z)
(x*y + x*z + y*z)/(x*y*z)

>>> together(1/(x*y) + 1/y**2)
(x + y)/(x*y**2)

>>> together(1/(1 + 1/x) + 1/(1 + 1/y))
(x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1))

>>> together(exp(1/x + 1/y))
exp(1/y + 1/x)
>>> together(exp(1/x + 1/y), deep=True)
exp((x + y)/(x*y))

>>> together(1/exp(x) + 1/(x*exp(x)))
(x + 1)*exp(-x)/x

>>> together(1/exp(2*x) + 1/(x*exp(3*x)))
(x*exp(x) + 1)*exp(-3*x)/x

Partial fraction decomposition

sympy.polys.partfrac.apart(f, x=None, full=False, **options)

Compute partial fraction decomposition of a rational function.

Given a rational function f, computes the partial fraction decomposition of f. Two algorithms are available: One is based on the undertermined coefficients method, the other is Bronstein’s full partial fraction decomposition algorithm.

The undetermined coefficients method (selected by full=False) uses polynomial factorization (and therefore accepts the same options as factor) for the denominator. Per default it works over the rational numbers, therefore decomposition of denominators with non-rational roots (e.g. irrational, complex roots) is not supported by default (see options of factor).

Bronstein’s algorithm can be selected by using full=True and allows a decomposition of denominators with non-rational roots. A human-readable result can be obtained via doit() (see examples below).

Examples

>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y

By default, using the undetermined coefficients method:

>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)

The undetermined coefficients method does not provide a result when the denominators roots are not rational:

>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)

You can choose Bronstein’s algorithm by setting full=True:

>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))

Calling doit() yields a human-readable result:

>>> apart(y/(x**2 + x + 1), x, full=True).doit()
(-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 -
    2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2)

See also

apart_list, assemble_partfrac_list

sympy.polys.partfrac.apart_list(f, x=None, dummies=None, **options)

Compute partial fraction decomposition of a rational function and return the result in structured form.

Given a rational function f compute the partial fraction decomposition of f. Only Bronstein’s full partial fraction decomposition algorithm is supported by this method. The return value is highly structured and perfectly suited for further algorithmic treatment rather than being human-readable. The function returns a tuple holding three elements:

• The first item is the common coefficient, free of the variable \(x\) used for decomposition. (It is an element of the base field \(K\).)

• The second item is the polynomial part of the decomposition. This can be the zero polynomial. (It is an element of \(K[x]\).)

• The third part itself is a list of quadruples. Each quadruple has the following elements in this order:

  • The (not necessarily irreducible) polynomial \(D\) whose roots \(w_i\) appear in the linear denominator of a bunch of related fraction terms. (This item can also be a list of explicit roots. However, at the moment apart_list never returns a result this way, but the related assemble_partfrac_list function accepts this format as input.)

  • The numerator of the fraction, written as a function of the root \(w\)

  • The linear denominator of the fraction excluding its power exponent, written as a function of the root \(w\).

  • The power to which the denominator has to be raised.

On can always rebuild a plain expression by using the function assemble_partfrac_list.

Examples

A first example:

>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, t

>>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(2*x + 4, x, domain='ZZ'),
[(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)])

>>> assemble_partfrac_list(pfd)
2*x + 4 + 4/(x - 1)

Second example:

>>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
>>> pfd = apart_list(f)
>>> pfd
(-1,
Poly(2/3, x, domain='QQ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])

>>> assemble_partfrac_list(pfd)
-2/3 - 2/(x - 2)

Another example, showing symbolic parameters:

>>> pfd = apart_list(t/(x**2 + x + t), x)
>>> pfd
(1,
Poly(0, x, domain='ZZ[t]'),
[(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'),
Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)),
Lambda(_a, -_a + x),
1)])

>>> assemble_partfrac_list(pfd)
RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x)))

This example is taken from Bronstein’s original paper:

>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])

>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)

See also

apart, assemble_partfrac_list

References

R559

[Bronstein93]

sympy.polys.partfrac.assemble_partfrac_list(partial_list)

Reassemble a full partial fraction decomposition from a structured result obtained by the function apart_list.

Examples

This example is taken from Bronstein’s original paper:

>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, y

>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])

>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)

If we happen to know some roots we can provide them easily inside the structure:

>>> pfd = apart_list(2/(x**2-2))
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w**2 - 2, _w, domain='ZZ'),
Lambda(_a, _a/2),
Lambda(_a, -_a + x),
1)])

>>> pfda = assemble_partfrac_list(pfd)
>>> pfda
RootSum(_w**2 - 2, Lambda(_a, _a/(-_a + x)))/2

>>> pfda.doit()
-sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2)))

>>> from sympy import Dummy, Poly, Lambda, sqrt
>>> a = Dummy("a")
>>> pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)])

>>> assemble_partfrac_list(pfd)
-sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2)))

See also

apart, apart_list

Dispersion of Polynomials

sympy.polys.dispersion.dispersionset(p, q=None, *gens, **args)

Compute the dispersion set of two polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion set \(\operatorname{J}(f, g)\) is defined as:

\[\begin{split}\operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}\end{split}\]

For a single polynomial one defines \(\operatorname{J}(f) := \operatorname{J}(f, f)\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersion

References

R560

[ManWright94]

R561

[Koepf98]

R562

[Abramov71]

R563

[Man93]

sympy.polys.dispersion.dispersion(p, q=None, *gens, **args)

Compute the dispersion of polynomials.

For two polynomials \(f(x)\) and \(g(x)\) with \(\deg f > 0\) and \(\deg g > 0\) the dispersion \(\operatorname{dis}(f, g)\) is defined as:

\[\begin{split}\operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \}\end{split}\]

and for a single polynomial \(\operatorname{dis}(f) := \operatorname{dis}(f, f)\). Note that we make the definition \(\max\{\} := -\infty\).

Examples

>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x

Dispersion set and dispersion of a simple polynomial:

>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6

Note that the definition of the dispersion is not symmetric:

>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo

The maximum of an empty set is defined to be \(-\infty\) as seen in this example.

Computing the dispersion also works over field extensions:

>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]

We can even perform the computations for polynomials having symbolic coefficients:

>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]

See also

dispersionset

References

R564

[ManWright94]

R565

[Koepf98]

R566

[Abramov71]

R567

[Man93]