Elements of Laurent polynomial rings¶

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial[source]¶

Bases: CommutativeAlgebraElement

Base class for Laurent polynomials.

Return a copy of this Laurent polynomial, with coefficients in R.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: a = x^2 + 3*x^3 + 5*x^-1
sage: a.change_ring(GF(3))
2*x^-1 + x^2

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> a = x**Integer(2) + Integer(3)*x**Integer(3) + Integer(5)*x**-Integer(1)
>>> a.change_ring(GF(Integer(3)))
2*x^-1 + x^2

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
a = x^2 + 3*x^3 + 5*x^-1
a.change_ring(GF(3))

Check that Issue #22277 is fixed:

Sage

sage: # needs sage.modules
sage: R.<x, y> = LaurentPolynomialRing(QQ)
sage: a = 2*x^2 + 3*x^3 + 4*x^-1
sage: a.change_ring(GF(3))
-x^2 + x^-1

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> R = LaurentPolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> a = Integer(2)*x**Integer(2) + Integer(3)*x**Integer(3) + Integer(4)*x**-Integer(1)
>>> a.change_ring(GF(Integer(3)))
-x^2 + x^-1

Sage Live

# needs sage.modules
R.<x, y> = LaurentPolynomialRing(QQ)
a = 2*x^2 + 3*x^3 + 4*x^-1
a.change_ring(GF(3))

dict()[source]¶

Abstract dict method.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
sage: LaurentPolynomial.monomial_coefficients(x)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
>>> LaurentPolynomial.monomial_coefficients(x)
Traceback (most recent call last):
...
NotImplementedError

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
LaurentPolynomial.monomial_coefficients(x)

hamming_weight()[source]¶

Return the hamming weight of self.

The hamming weight is number of nonzero coefficients and also known as the weight or sparsity.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - 1
sage: f.hamming_weight()
2

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = x**Integer(3) - Integer(1)
>>> f.hamming_weight()
2

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
f = x^3 - 1
f.hamming_weight()

map_coefficients(f, new_base_ring=None)[source]¶

Apply f to the coefficients of self.

If f is a sage.categories.map.Map, then the resulting polynomial will be defined over the codomain of f. Otherwise, the resulting polynomial will be over the same ring as self. Set new_base_ring to override this behavior.

INPUT:

f – a callable that will be applied to the coefficients of self
new_base_ring – (optional) if given, the resulting polynomial will be defined over this ring

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(9)
sage: R.<x> = LaurentPolynomialRing(k)
sage: f = x*a + a
sage: f.map_coefficients(lambda a: a + 1)
(a + 1) + (a + 1)*x
sage: R.<x,y> = LaurentPolynomialRing(k, 2)                                 # needs sage.modules
sage: f = x*a + 2*x^3*y*a + a                                               # needs sage.modules
sage: f.map_coefficients(lambda a: a + 1)                                   # needs sage.modules
(2*a + 1)*x^3*y + (a + 1)*x + a + 1

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(9), names=('a',)); (a,) = k._first_ngens(1)
>>> R = LaurentPolynomialRing(k, names=('x',)); (x,) = R._first_ngens(1)
>>> f = x*a + a
>>> f.map_coefficients(lambda a: a + Integer(1))
(a + 1) + (a + 1)*x
>>> R = LaurentPolynomialRing(k, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)# needs sage.modules
>>> f = x*a + Integer(2)*x**Integer(3)*y*a + a                                               # needs sage.modules
>>> f.map_coefficients(lambda a: a + Integer(1))                                   # needs sage.modules
(2*a + 1)*x^3*y + (a + 1)*x + a + 1

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(9)
R.<x> = LaurentPolynomialRing(k)
f = x*a + a
f.map_coefficients(lambda a: a + 1)
R.<x,y> = LaurentPolynomialRing(k, 2)                                 # needs sage.modules
f = x*a + 2*x^3*y*a + a                                               # needs sage.modules
f.map_coefficients(lambda a: a + 1)                                   # needs sage.modules

Examples with different base ring:

Sage

sage: # needs sage.modules sage.rings.finite_rings
sage: R.<r> = GF(9); S.<s> = GF(81)
sage: h = Hom(R, S)[0]; h
Ring morphism:
  From: Finite Field in r of size 3^2
  To:   Finite Field in s of size 3^4
  Defn: r |--> 2*s^3 + 2*s^2 + 1
sage: T.<X,Y> = LaurentPolynomialRing(R, 2)
sage: f = r*X + Y
sage: g = f.map_coefficients(h); g
(2*s^3 + 2*s^2 + 1)*X + Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y
 over Finite Field in s of size 3^4
sage: h = lambda x: x.trace()
sage: g = f.map_coefficients(h); g
X - Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y
 over Finite Field in r of size 3^2
sage: g = f.map_coefficients(h, new_base_ring=GF(3)); g
X - Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y over Finite Field of size 3

Python

>>> from sage.all import *
>>> # needs sage.modules sage.rings.finite_rings
>>> R = GF(Integer(9), names=('r',)); (r,) = R._first_ngens(1); S = GF(Integer(81), names=('s',)); (s,) = S._first_ngens(1)
>>> h = Hom(R, S)[Integer(0)]; h
Ring morphism:
  From: Finite Field in r of size 3^2
  To:   Finite Field in s of size 3^4
  Defn: r |--> 2*s^3 + 2*s^2 + 1
>>> T = LaurentPolynomialRing(R, Integer(2), names=('X', 'Y',)); (X, Y,) = T._first_ngens(2)
>>> f = r*X + Y
>>> g = f.map_coefficients(h); g
(2*s^3 + 2*s^2 + 1)*X + Y
>>> g.parent()
Multivariate Laurent Polynomial Ring in X, Y
 over Finite Field in s of size 3^4
>>> h = lambda x: x.trace()
>>> g = f.map_coefficients(h); g
X - Y
>>> g.parent()
Multivariate Laurent Polynomial Ring in X, Y
 over Finite Field in r of size 3^2
>>> g = f.map_coefficients(h, new_base_ring=GF(Integer(3))); g
X - Y
>>> g.parent()
Multivariate Laurent Polynomial Ring in X, Y over Finite Field of size 3

Sage Live

# needs sage.modules sage.rings.finite_rings
R.<r> = GF(9); S.<s> = GF(81)
h = Hom(R, S)[0]; h
T.<X,Y> = LaurentPolynomialRing(R, 2)
f = r*X + Y
g = f.map_coefficients(h); g
g.parent()
h = lambda x: x.trace()
g = f.map_coefficients(h); g
g.parent()
g = f.map_coefficients(h, new_base_ring=GF(3)); g
g.parent()

monomial_coefficients()[source]¶

Abstract dict method.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
sage: LaurentPolynomial.monomial_coefficients(x)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
>>> LaurentPolynomial.monomial_coefficients(x)
Traceback (most recent call last):
...
NotImplementedError

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
LaurentPolynomial.monomial_coefficients(x)

number_of_terms()[source]¶

Abstract method for number of terms

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
sage: LaurentPolynomial.number_of_terms(x)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
>>> LaurentPolynomial.number_of_terms(x)
Traceback (most recent call last):
...
NotImplementedError

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
LaurentPolynomial.number_of_terms(x)

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_univariate[source]¶

Bases: LaurentPolynomial

A univariate Laurent polynomial in the form of \(t^n \cdot f\) where \(f\) is a polynomial in \(t\).

INPUT:

parent – a Laurent polynomial ring
f – a polynomial (or something that can be coerced to one)
n – integer (default: 0)

AUTHORS:

Tom Boothby (2011) copied this class almost verbatim from laurent_series_ring_element.pyx, so most of the credit goes to William Stein, David Joyner, and Robert Bradshaw
Travis Scrimshaw (09-2013): Cleaned-up and added a few extra methods

coefficients()[source]¶

Return the nonzero coefficients of self.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.coefficients()
[-5, 1, 1, -10/3]

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = -Integer(5)/t**(Integer(2)) + t + t**Integer(2) - Integer(10)/Integer(3)*t**Integer(3)
>>> f.coefficients()
[-5, 1, 1, -10/3]

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
f = -5/t^(2) + t + t^2 - 10/3*t^3
f.coefficients()

constant_coefficient()[source]¶

Return the coefficient of the constant term of self.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = 3*t^-2 - t^-1 + 3 + t^2
sage: f.constant_coefficient()
3
sage: g = -2*t^-2 + t^-1 + 3*t
sage: g.constant_coefficient()
0

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(3)*t**-Integer(2) - t**-Integer(1) + Integer(3) + t**Integer(2)
>>> f.constant_coefficient()
3
>>> g = -Integer(2)*t**-Integer(2) + t**-Integer(1) + Integer(3)*t
>>> g.constant_coefficient()
0

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
f = 3*t^-2 - t^-1 + 3 + t^2
f.constant_coefficient()
g = -2*t^-2 + t^-1 + 3*t
g.constant_coefficient()

degree()[source]¶

Return the degree of self.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: g = x^2 - x^4
sage: g.degree()
4
sage: g = -10/x^5 + x^2 - x^7
sage: g.degree()
7

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> g = x**Integer(2) - x**Integer(4)
>>> g.degree()
4
>>> g = -Integer(10)/x**Integer(5) + x**Integer(2) - x**Integer(7)
>>> g.degree()
7

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
g = x^2 - x^4
g.degree()
g = -10/x^5 + x^2 - x^7
g.degree()

The zero polynomial is defined to have degree \(-\infty\):

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: R.zero().degree()
-Infinity

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> R.zero().degree()
-Infinity

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
R.zero().degree()

derivative(*args)[source]¶

The formal derivative of this Laurent polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied. See documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: g = 1/x^10 - x + x^2 - x^4
sage: g.derivative()
-10*x^-11 - 1 + 2*x - 4*x^3
sage: g.derivative(x)
-10*x^-11 - 1 + 2*x - 4*x^3

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> g = Integer(1)/x**Integer(10) - x + x**Integer(2) - x**Integer(4)
>>> g.derivative()
-10*x^-11 - 1 + 2*x - 4*x^3
>>> g.derivative(x)
-10*x^-11 - 1 + 2*x - 4*x^3

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
g = 1/x^10 - x + x^2 - x^4
g.derivative()
g.derivative(x)

Sage

sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = LaurentPolynomialRing(R)
sage: f = 2*t/x + (3*t^2 + 6*t)*x
sage: f.derivative()
-2*t*x^-2 + (3*t^2 + 6*t)
sage: f.derivative(x)
-2*t*x^-2 + (3*t^2 + 6*t)
sage: f.derivative(t)
2*x^-1 + (6*t + 6)*x

Python

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> S = LaurentPolynomialRing(R, names=('x',)); (x,) = S._first_ngens(1)
>>> f = Integer(2)*t/x + (Integer(3)*t**Integer(2) + Integer(6)*t)*x
>>> f.derivative()
-2*t*x^-2 + (3*t^2 + 6*t)
>>> f.derivative(x)
-2*t*x^-2 + (3*t^2 + 6*t)
>>> f.derivative(t)
2*x^-1 + (6*t + 6)*x

Sage Live

R.<t> = PolynomialRing(ZZ)
S.<x> = LaurentPolynomialRing(R)
f = 2*t/x + (3*t^2 + 6*t)*x
f.derivative()
f.derivative(x)
f.derivative(t)

dict()[source]¶

Return a dictionary representing self.

EXAMPLES:

Sage

sage: R.<x,y> = ZZ[]
sage: Q.<t> = LaurentPolynomialRing(R)
sage: f = (x^3 + y/t^3)^3 + t^2; f
y^3*t^-9 + 3*x^3*y^2*t^-6 + 3*x^6*y*t^-3 + x^9 + t^2
sage: f.monomial_coefficients()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Python

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> Q = LaurentPolynomialRing(R, names=('t',)); (t,) = Q._first_ngens(1)
>>> f = (x**Integer(3) + y/t**Integer(3))**Integer(3) + t**Integer(2); f
y^3*t^-9 + 3*x^3*y^2*t^-6 + 3*x^6*y*t^-3 + x^9 + t^2
>>> f.monomial_coefficients()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Sage Live

R.<x,y> = ZZ[]
Q.<t> = LaurentPolynomialRing(R)
f = (x^3 + y/t^3)^3 + t^2; f
f.monomial_coefficients()

dict is an alias:

Sage

sage: f.dict()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Python

>>> from sage.all import *
>>> f.dict()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Sage Live

f.dict()

divides(other)[source]¶

Return True if self divides other.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: (2*x**-1 + 1).divides(4*x**-2 - 1)
True
sage: (2*x + 1).divides(4*x**2 + 1)
False
sage: (2*x + x**-1).divides(R(0))
True
sage: R(0).divides(2*x ** -1 + 1)
False
sage: R(0).divides(R(0))
True
sage: R.<x> = LaurentPolynomialRing(Zmod(6))
sage: p = 4*x + 3*x^-1
sage: q = 5*x^2 + x + 2*x^-2
sage: p.divides(q)
False

sage: R.<x,y> = GF(2)[]
sage: S.<z> = LaurentPolynomialRing(R)
sage: p = (x+y+1) * z**-1 + x*y
sage: q = (y^2-x^2) * z**-2 + z + x-y
sage: p.divides(q), p.divides(p*q)                                          # needs sage.libs.singular
(False, True)

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(2)*x**-Integer(1) + Integer(1)).divides(Integer(4)*x**-Integer(2) - Integer(1))
True
>>> (Integer(2)*x + Integer(1)).divides(Integer(4)*x**Integer(2) + Integer(1))
False
>>> (Integer(2)*x + x**-Integer(1)).divides(R(Integer(0)))
True
>>> R(Integer(0)).divides(Integer(2)*x ** -Integer(1) + Integer(1))
False
>>> R(Integer(0)).divides(R(Integer(0)))
True
>>> R = LaurentPolynomialRing(Zmod(Integer(6)), names=('x',)); (x,) = R._first_ngens(1)
>>> p = Integer(4)*x + Integer(3)*x**-Integer(1)
>>> q = Integer(5)*x**Integer(2) + x + Integer(2)*x**-Integer(2)
>>> p.divides(q)
False

>>> R = GF(Integer(2))['x, y']; (x, y,) = R._first_ngens(2)
>>> S = LaurentPolynomialRing(R, names=('z',)); (z,) = S._first_ngens(1)
>>> p = (x+y+Integer(1)) * z**-Integer(1) + x*y
>>> q = (y**Integer(2)-x**Integer(2)) * z**-Integer(2) + z + x-y
>>> p.divides(q), p.divides(p*q)                                          # needs sage.libs.singular
(False, True)

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
(2*x**-1 + 1).divides(4*x**-2 - 1)
(2*x + 1).divides(4*x**2 + 1)
(2*x + x**-1).divides(R(0))
R(0).divides(2*x ** -1 + 1)
R(0).divides(R(0))
R.<x> = LaurentPolynomialRing(Zmod(6))
p = 4*x + 3*x^-1
q = 5*x^2 + x + 2*x^-2
p.divides(q)
R.<x,y> = GF(2)[]
S.<z> = LaurentPolynomialRing(R)
p = (x+y+1) * z**-1 + x*y
q = (y^2-x^2) * z**-2 + z + x-y
p.divides(q), p.divides(p*q)                                          # needs sage.libs.singular

euclidean_degree()[source]¶

Return the degree of self as an element of an Euclidean domain.

This is the Euclidean degree of the underlying polynomial.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: (x^-5 + x^2).euclidean_degree()
7

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: (x^-5 + x^2).euclidean_degree()
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> (x**-Integer(5) + x**Integer(2)).euclidean_degree()
7

>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> (x**-Integer(5) + x**Integer(2)).euclidean_degree()
Traceback (most recent call last):
...
NotImplementedError

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
(x^-5 + x^2).euclidean_degree()
R.<x> = LaurentPolynomialRing(ZZ)
(x^-5 + x^2).euclidean_degree()

exponents()[source]¶

Return the exponents appearing in self with nonzero coefficients.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.exponents()
[-2, 1, 2, 3]

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = -Integer(5)/t**(Integer(2)) + t + t**Integer(2) - Integer(10)/Integer(3)*t**Integer(3)
>>> f.exponents()
[-2, 1, 2, 3]

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
f = -5/t^(2) + t + t^2 - 10/3*t^3
f.exponents()

factor()[source]¶

Return a Laurent monomial (the unit part of the factorization) and a factored polynomial.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(ZZ)
sage: f = 4*t^-7 + 3*t^3 + 2*t^4 + t^-6
sage: f.factor()                                                            # needs sage.libs.pari
(t^-7) * (4 + t + 3*t^10 + 2*t^11)

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(4)*t**-Integer(7) + Integer(3)*t**Integer(3) + Integer(2)*t**Integer(4) + t**-Integer(6)
>>> f.factor()                                                            # needs sage.libs.pari
(t^-7) * (4 + t + 3*t^10 + 2*t^11)

Sage Live

R.<t> = LaurentPolynomialRing(ZZ)
f = 4*t^-7 + 3*t^3 + 2*t^4 + t^-6
f.factor()                                                            # needs sage.libs.pari

gcd(right)[source]¶

Return the gcd of self with right where the common divisor d makes both self and right into polynomials with the lowest possible degree.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: t.gcd(2)
1
sage: gcd(t^-2 + 1, t^-4 + 3*t^-1)
t^-4
sage: gcd((t^-2 + t)*(t + t^-1), (t^5 + t^8)*(1 + t^-2))
t^-3 + t^-1 + 1 + t^2

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> t.gcd(Integer(2))
1
>>> gcd(t**-Integer(2) + Integer(1), t**-Integer(4) + Integer(3)*t**-Integer(1))
t^-4
>>> gcd((t**-Integer(2) + t)*(t + t**-Integer(1)), (t**Integer(5) + t**Integer(8))*(Integer(1) + t**-Integer(2)))
t^-3 + t^-1 + 1 + t^2

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
t.gcd(2)
gcd(t^-2 + 1, t^-4 + 3*t^-1)
gcd((t^-2 + t)*(t + t^-1), (t^5 + t^8)*(1 + t^-2))

integral()[source]¶

The formal integral of this Laurent series with 0 constant term.

EXAMPLES:

The integral may or may not be defined if the base ring is not a field.

Sage

sage: t = LaurentPolynomialRing(ZZ, 't').0
sage: f = 2*t^-3 + 3*t^2
sage: f.integral()
-t^-2 + t^3

Python

>>> from sage.all import *
>>> t = LaurentPolynomialRing(ZZ, 't').gen(0)
>>> f = Integer(2)*t**-Integer(3) + Integer(3)*t**Integer(2)
>>> f.integral()
-t^-2 + t^3

Sage Live

t = LaurentPolynomialRing(ZZ, 't').0
f = 2*t^-3 + 3*t^2
f.integral()

Sage

sage: f = t^3
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: coefficients of integral cannot be coerced into the base ring

Python

>>> from sage.all import *
>>> f = t**Integer(3)
>>> f.integral()
Traceback (most recent call last):
...
ArithmeticError: coefficients of integral cannot be coerced into the base ring

Sage Live

f = t^3
f.integral()

The integral of \(1/t\) is \(\log(t)\), which is not given by a Laurent polynomial:

Sage

sage: t = LaurentPolynomialRing(ZZ,'t').0
sage: f = -1/t^3 - 31/t
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: the integral of is not a Laurent polynomial, since t^-1 has nonzero coefficient

Python

>>> from sage.all import *
>>> t = LaurentPolynomialRing(ZZ,'t').gen(0)
>>> f = -Integer(1)/t**Integer(3) - Integer(31)/t
>>> f.integral()
Traceback (most recent call last):
...
ArithmeticError: the integral of is not a Laurent polynomial, since t^-1 has nonzero coefficient

Sage Live

t = LaurentPolynomialRing(ZZ,'t').0
f = -1/t^3 - 31/t
f.integral()

Another example with just one negative coefficient:

Sage

sage: A.<t> = LaurentPolynomialRing(QQ)
sage: f = -2*t^(-4)
sage: f.integral()
2/3*t^-3
sage: f.integral().derivative() == f
True

Python

>>> from sage.all import *
>>> A = LaurentPolynomialRing(QQ, names=('t',)); (t,) = A._first_ngens(1)
>>> f = -Integer(2)*t**(-Integer(4))
>>> f.integral()
2/3*t^-3
>>> f.integral().derivative() == f
True

Sage Live

A.<t> = LaurentPolynomialRing(QQ)
f = -2*t^(-4)
f.integral()
f.integral().derivative() == f

inverse_mod(a, m)[source]¶

Invert the polynomial a with respect to m, or raise a ValueError if no such inverse exists.

The parameter m may be either a single polynomial or an ideal (for consistency with inverse_mod() in other rings).

ALGORITHM: Solve the system \(as + mt = 1\), returning \(s\) as the inverse of \(a\) mod \(m\).

EXAMPLES:

Sage

sage: S.<t> = LaurentPolynomialRing(QQ)
sage: f = inverse_mod(t^-2 + 1, t^-3 + 1); f
1/2*t^2 - 1/2*t^3 - 1/2*t^4
sage: f * (t^-2 + 1) + (1/2*t^4 + 1/2*t^3) * (t^-3 + 1)
1

Python

>>> from sage.all import *
>>> S = LaurentPolynomialRing(QQ, names=('t',)); (t,) = S._first_ngens(1)
>>> f = inverse_mod(t**-Integer(2) + Integer(1), t**-Integer(3) + Integer(1)); f
1/2*t^2 - 1/2*t^3 - 1/2*t^4
>>> f * (t**-Integer(2) + Integer(1)) + (Integer(1)/Integer(2)*t**Integer(4) + Integer(1)/Integer(2)*t**Integer(3)) * (t**-Integer(3) + Integer(1))
1

Sage Live

S.<t> = LaurentPolynomialRing(QQ)
f = inverse_mod(t^-2 + 1, t^-3 + 1); f
f * (t^-2 + 1) + (1/2*t^4 + 1/2*t^3) * (t^-3 + 1)

inverse_of_unit()[source]¶

Return the inverse of self if a unit.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (t^-2).inverse_of_unit()
t^2
sage: (t + 2).inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: element is not a unit

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> (t**-Integer(2)).inverse_of_unit()
t^2
>>> (t + Integer(2)).inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: element is not a unit

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
(t^-2).inverse_of_unit()
(t + 2).inverse_of_unit()

is_constant()[source]¶

Return whether this Laurent polynomial is constant.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: x.is_constant()
False
sage: R.one().is_constant()
True
sage: (x^-2).is_constant()
False
sage: (x^2).is_constant()
False
sage: (x^-2 + 2).is_constant()
False
sage: R(0).is_constant()
True
sage: R(42).is_constant()
True
sage: x.is_constant()
False
sage: (1/x).is_constant()
False

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> x.is_constant()
False
>>> R.one().is_constant()
True
>>> (x**-Integer(2)).is_constant()
False
>>> (x**Integer(2)).is_constant()
False
>>> (x**-Integer(2) + Integer(2)).is_constant()
False
>>> R(Integer(0)).is_constant()
True
>>> R(Integer(42)).is_constant()
True
>>> x.is_constant()
False
>>> (Integer(1)/x).is_constant()
False

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
x.is_constant()
R.one().is_constant()
(x^-2).is_constant()
(x^2).is_constant()
(x^-2 + 2).is_constant()
R(0).is_constant()
R(42).is_constant()
x.is_constant()
(1/x).is_constant()

is_monomial()[source]¶

Return True if self is a monomial; that is, if self is \(x^n\) for some integer \(n\).

EXAMPLES:

Sage

sage: k.<z> = LaurentPolynomialRing(QQ)
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^-2909).is_monomial()
True
sage: (38*z^-2909).is_monomial()
False

Python

>>> from sage.all import *
>>> k = LaurentPolynomialRing(QQ, names=('z',)); (z,) = k._first_ngens(1)
>>> z.is_monomial()
True
>>> k(Integer(1)).is_monomial()
True
>>> (z+Integer(1)).is_monomial()
False
>>> (z**-Integer(2909)).is_monomial()
True
>>> (Integer(38)*z**-Integer(2909)).is_monomial()
False

Sage Live

k.<z> = LaurentPolynomialRing(QQ)
z.is_monomial()
k(1).is_monomial()
(z+1).is_monomial()
(z^-2909).is_monomial()
(38*z^-2909).is_monomial()

is_square(root=False)[source]¶

Return whether this Laurent polynomial is a square.

If root is set to True then return a pair made of the boolean answer together with None or a square root.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)

sage: R.one().is_square()
True
sage: R(2).is_square()
False

sage: t.is_square()
False
sage: (t**-2).is_square()
True

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)

>>> R.one().is_square()
True
>>> R(Integer(2)).is_square()
False

>>> t.is_square()
False
>>> (t**-Integer(2)).is_square()
True

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
R.one().is_square()
R(2).is_square()
t.is_square()
(t**-2).is_square()

Usage of the root option:

Sage

sage: p = (1 + t^-1 - 2*t^3)
sage: p.is_square(root=True)
(False, None)
sage: (p**2).is_square(root=True)
(True, -t^-1 - 1 + 2*t^3)

Python

>>> from sage.all import *
>>> p = (Integer(1) + t**-Integer(1) - Integer(2)*t**Integer(3))
>>> p.is_square(root=True)
(False, None)
>>> (p**Integer(2)).is_square(root=True)
(True, -t^-1 - 1 + 2*t^3)

Sage Live

p = (1 + t^-1 - 2*t^3)
p.is_square(root=True)
(p**2).is_square(root=True)

The answer is dependent of the base ring:

Sage

sage: # needs sage.rings.number_field
sage: S.<u> = LaurentPolynomialRing(QQbar)
sage: (2 + 4*t + 2*t^2).is_square()
False
sage: (2 + 4*u + 2*u^2).is_square()
True

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> S = LaurentPolynomialRing(QQbar, names=('u',)); (u,) = S._first_ngens(1)
>>> (Integer(2) + Integer(4)*t + Integer(2)*t**Integer(2)).is_square()
False
>>> (Integer(2) + Integer(4)*u + Integer(2)*u**Integer(2)).is_square()
True

Sage Live

# needs sage.rings.number_field
S.<u> = LaurentPolynomialRing(QQbar)
(2 + 4*t + 2*t^2).is_square()
(2 + 4*u + 2*u^2).is_square()

is_unit()[source]¶

Return True if this Laurent polynomial is a unit in this ring.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (2 + t).is_unit()
False
sage: f = 2*t
sage: f.is_unit()
True
sage: 1/f
1/2*t^-1
sage: R(0).is_unit()
False
sage: R.<s> = LaurentPolynomialRing(ZZ)
sage: g = 2*s
sage: g.is_unit()
False
sage: 1/g
1/2*s^-1

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> (Integer(2) + t).is_unit()
False
>>> f = Integer(2)*t
>>> f.is_unit()
True
>>> Integer(1)/f
1/2*t^-1
>>> R(Integer(0)).is_unit()
False
>>> R = LaurentPolynomialRing(ZZ, names=('s',)); (s,) = R._first_ngens(1)
>>> g = Integer(2)*s
>>> g.is_unit()
False
>>> Integer(1)/g
1/2*s^-1

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
(2 + t).is_unit()
f = 2*t
f.is_unit()
1/f
R(0).is_unit()
R.<s> = LaurentPolynomialRing(ZZ)
g = 2*s
g.is_unit()
1/g

ALGORITHM: A Laurent polynomial is a unit if and only if its “unit part” is a unit.

is_zero()[source]¶

Return 1 if self is 0, else return 0.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x + x^2 + 3*x^4
sage: f.is_zero()
0
sage: z = 0*f
sage: z.is_zero()
1

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> f.is_zero()
0
>>> z = Integer(0)*f
>>> z.is_zero()
1

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x + x + x^2 + 3*x^4
f.is_zero()
z = 0*f
z.is_zero()

monomial_coefficients()[source]¶

Return a dictionary representing self.

EXAMPLES:

Sage

sage: R.<x,y> = ZZ[]
sage: Q.<t> = LaurentPolynomialRing(R)
sage: f = (x^3 + y/t^3)^3 + t^2; f
y^3*t^-9 + 3*x^3*y^2*t^-6 + 3*x^6*y*t^-3 + x^9 + t^2
sage: f.monomial_coefficients()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Python

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> Q = LaurentPolynomialRing(R, names=('t',)); (t,) = Q._first_ngens(1)
>>> f = (x**Integer(3) + y/t**Integer(3))**Integer(3) + t**Integer(2); f
y^3*t^-9 + 3*x^3*y^2*t^-6 + 3*x^6*y*t^-3 + x^9 + t^2
>>> f.monomial_coefficients()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Sage Live

R.<x,y> = ZZ[]
Q.<t> = LaurentPolynomialRing(R)
f = (x^3 + y/t^3)^3 + t^2; f
f.monomial_coefficients()

dict is an alias:

Sage

sage: f.dict()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Python

>>> from sage.all import *
>>> f.dict()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

Sage Live

f.dict()

monomial_reduction()[source]¶

Return the decomposition as a polynomial and a power of the variable. Constructed for compatibility with the multivariate case.

OUTPUT:

A tuple (u, t^n) where u is the underlying polynomial and n is the power of the exponent shift.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.monomial_reduction()
(3*x^5 + x^3 + 1, x^-1)

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> f.monomial_reduction()
(3*x^5 + x^3 + 1, x^-1)

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x + x^2 + 3*x^4
f.monomial_reduction()

number_of_terms()[source]¶

Return the number of nonzero coefficients of self.

Also called weight, hamming weight or sparsity.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - 1
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+1)^100
sage: f.number_of_terms()
101

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = x**Integer(3) - Integer(1)
>>> f.number_of_terms()
2
>>> R(Integer(0)).number_of_terms()
0
>>> f = (x+Integer(1))**Integer(100)
>>> f.number_of_terms()
101

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
f = x^3 - 1
f.number_of_terms()
R(0).number_of_terms()
f = (x+1)^100
f.number_of_terms()

The method hamming_weight() is an alias:

Sage

sage: f.hamming_weight()
101

Python

>>> from sage.all import *
>>> f.hamming_weight()
101

Sage Live

f.hamming_weight()

polynomial_construction()[source]¶

Return the polynomial and the shift in power used to construct the Laurent polynomial \(t^n u\).

OUTPUT:

A tuple (u, n) where u is the underlying polynomial and n is the power of the exponent shift.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.polynomial_construction()
(3*x^5 + x^3 + 1, -1)

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> f.polynomial_construction()
(3*x^5 + x^3 + 1, -1)

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x + x^2 + 3*x^4
f.polynomial_construction()

quo_rem(other)[source]¶

Divide self by other and return a quotient q and a remainder r such that self == q * other + r.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (t^-3 - t^3).quo_rem(t^-1 - t)
(t^-2 + 1 + t^2, 0)
sage: (t^-2 + 3 + t).quo_rem(t^-4)
(t^2 + 3*t^4 + t^5, 0)

sage: num = t^-2 + t
sage: den = t^-2 + 1
sage: q, r = num.quo_rem(den)
sage: num == q * den + r
True

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> (t**-Integer(3) - t**Integer(3)).quo_rem(t**-Integer(1) - t)
(t^-2 + 1 + t^2, 0)
>>> (t**-Integer(2) + Integer(3) + t).quo_rem(t**-Integer(4))
(t^2 + 3*t^4 + t^5, 0)

>>> num = t**-Integer(2) + t
>>> den = t**-Integer(2) + Integer(1)
>>> q, r = num.quo_rem(den)
>>> num == q * den + r
True

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
(t^-3 - t^3).quo_rem(t^-1 - t)
(t^-2 + 3 + t).quo_rem(t^-4)
num = t^-2 + t
den = t^-2 + 1
q, r = num.quo_rem(den)
num == q * den + r

residue()[source]¶

Return the residue of self.

The residue is the coefficient of \(t^-1\).

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = 3*t^-2 - t^-1 + 3 + t^2
sage: f.residue()
-1
sage: g = -2*t^-2 + 4 + 3*t
sage: g.residue()
0
sage: f.residue().parent()
Rational Field

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(3)*t**-Integer(2) - t**-Integer(1) + Integer(3) + t**Integer(2)
>>> f.residue()
-1
>>> g = -Integer(2)*t**-Integer(2) + Integer(4) + Integer(3)*t
>>> g.residue()
0
>>> f.residue().parent()
Rational Field

Sage Live

R.<t> = LaurentPolynomialRing(QQ)
f = 3*t^-2 - t^-1 + 3 + t^2
f.residue()
g = -2*t^-2 + 4 + 3*t
g.residue()
f.residue().parent()

shift(k)[source]¶

Return this Laurent polynomial multiplied by the power \(t^n\). Does not change this polynomial.

EXAMPLES:

Sage

sage: R.<t> = LaurentPolynomialRing(QQ['y'])
sage: f = (t+t^-1)^4; f
t^-4 + 4*t^-2 + 6 + 4*t^2 + t^4
sage: f.shift(10)
t^6 + 4*t^8 + 6*t^10 + 4*t^12 + t^14
sage: f >> 10
t^-14 + 4*t^-12 + 6*t^-10 + 4*t^-8 + t^-6
sage: f << 4
1 + 4*t^2 + 6*t^4 + 4*t^6 + t^8

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ['y'], names=('t',)); (t,) = R._first_ngens(1)
>>> f = (t+t**-Integer(1))**Integer(4); f
t^-4 + 4*t^-2 + 6 + 4*t^2 + t^4
>>> f.shift(Integer(10))
t^6 + 4*t^8 + 6*t^10 + 4*t^12 + t^14
>>> f >> Integer(10)
t^-14 + 4*t^-12 + 6*t^-10 + 4*t^-8 + t^-6
>>> f << Integer(4)
1 + 4*t^2 + 6*t^4 + 4*t^6 + t^8

Sage Live

R.<t> = LaurentPolynomialRing(QQ['y'])
f = (t+t^-1)^4; f
f.shift(10)
f >> 10
f << 4

truncate(n)[source]¶

Return a polynomial with degree at most \(n-1\) whose \(j\)-th coefficients agree with self for all \(j < n\).

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x^12 + x^3 + x^5 + x^9
sage: f.truncate(10)
x^-12 + x^3 + x^5 + x^9
sage: f.truncate(5)
x^-12 + x^3
sage: f.truncate(-16)
0

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x**Integer(12) + x**Integer(3) + x**Integer(5) + x**Integer(9)
>>> f.truncate(Integer(10))
x^-12 + x^3 + x^5 + x^9
>>> f.truncate(Integer(5))
x^-12 + x^3
>>> f.truncate(-Integer(16))
0

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x^12 + x^3 + x^5 + x^9
f.truncate(10)
f.truncate(5)
f.truncate(-16)

valuation(p=None)[source]¶

Return the valuation of self.

The valuation of a Laurent polynomial \(t^n u\) is \(n\) plus the valuation of \(u\).

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = 1/x + x^2 + 3*x^4
sage: g = 1 - x + x^2 - x^4
sage: f.valuation()
-1
sage: g.valuation()
0

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> g = Integer(1) - x + x**Integer(2) - x**Integer(4)
>>> f.valuation()
-1
>>> g.valuation()
0

Sage Live

R.<x> = LaurentPolynomialRing(ZZ)
f = 1/x + x^2 + 3*x^4
g = 1 - x + x^2 - x^4
f.valuation()
g.valuation()

variable_name()[source]¶

Return the name of variable of self as a string.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.variable_name()
'x'

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> f.variable_name()
'x'

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x + x^2 + 3*x^4
f.variable_name()

variables()[source]¶

Return the tuple of variables occurring in this Laurent polynomial.

EXAMPLES:

Sage

sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.variables()
(x,)
sage: R.one().variables()
()

Python

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = Integer(1)/x + x**Integer(2) + Integer(3)*x**Integer(4)
>>> f.variables()
(x,)
>>> R.one().variables()
()

Sage Live

R.<x> = LaurentPolynomialRing(QQ)
f = 1/x + x^2 + 3*x^4
f.variables()
R.one().variables()

xgcd(other)[source]¶

Extended gcd() for univariate Laurent polynomial rings over a field.

OUTPUT:

A triple (g, p, q) such that g is the gcd() of self (\(= a\)) and other (\(= b\)), and p and q are cofactors satisfying the Bezout identity

\[g = p \cdot a + q \cdot b.\]

EXAMPLES:

Sage

sage: S.<t> = LaurentPolynomialRing(QQ)
sage: a = t^-2 + 1
sage: b = t^-3 + 1
sage: g, p, q = a.xgcd(b); (g, p, q)
(t^-3, 1/2*t^-1 - 1/2 - 1/2*t, 1/2 + 1/2*t)
sage: g == p * a + q * b
True
sage: g == a.gcd(b)
True
sage: t.xgcd(t)
(t, 0, 1)
sage: t.xgcd(5)
(1, 0, 1/5)

Python

>>> from sage.all import *
>>> S = LaurentPolynomialRing(QQ, names=('t',)); (t,) = S._first_ngens(1)
>>> a = t**-Integer(2) + Integer(1)
>>> b = t**-Integer(3) + Integer(1)
>>> g, p, q = a.xgcd(b); (g, p, q)
(t^-3, 1/2*t^-1 - 1/2 - 1/2*t, 1/2 + 1/2*t)
>>> g == p * a + q * b
True
>>> g == a.gcd(b)
True
>>> t.xgcd(t)
(t, 0, 1)
>>> t.xgcd(Integer(5))
(1, 0, 1/5)

Sage Live

S.<t> = LaurentPolynomialRing(QQ)
a = t^-2 + 1
b = t^-3 + 1
g, p, q = a.xgcd(b); (g, p, q)
g == p * a + q * b
g == a.gcd(b)
t.xgcd(t)
t.xgcd(5)