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, *gens, **args)

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

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)
(-x/5 + 3/5, 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)
(-x/5 + 3/5, x**2/5 - 6*x/5 + 2, x + 1)

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

Invert f modulo g when possible.

See also

sympy.core.numbers.mod_inverse

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

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:

\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\} \}

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

See also

dispersionset

References

1. [ManWright94]
2. [Koepf98]
3. [Abramov71]
4. [Man93]

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]

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:

\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\}

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

See also

dispersion

References

1. [ManWright94]
2. [Koepf98]
3. [Abramov71]
4. [Man93]

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]

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.

See also

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

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

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.

Examples

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

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

>>> gff_list(f)
[(x, 1), (x + 2, 4)]

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

>>> f = 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

>>> gff_list(f)
[(x**3 + 7, 2), (x**2 + 5*x, 3)]

>>> ff(x**3 + 7, 2)*ff(x**2 + 5*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.

See also

sympy.ntheory.factor_.factorint

Examples

>>> from sympy import factor, sqrt
>>> 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)

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().

References

1. [Buchberger01]
2. [Cox97]

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')

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 polynomial expressions.

EC(f, 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(f, 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(f, 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(f, 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(f, order=None)

Returns the leading 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).LM()
x**2*y**0

LT(f, order=None)

Returns the leading 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).LT()
(x**2*y**0, 4)

TC(f)

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(f)

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(f, 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(f, 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(f)

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(f)

Returns all monomials from a univariate polynomial f.

See also

all_terms

Examples

>>> from sympy import Poly