Ring of Laurent Polynomials

If \(R\) is a commutative ring, then the ring of Laurent polynomials in \(n\) variables over \(R\) is \(R[x_1^{\pm 1}, x_2^{\pm 1}, \ldots, x_n^{\pm 1}]\). We implement it as a quotient ring

\[R[x_1, y_1, x_2, y_2, \ldots, x_n, y_n] / (x_1 y_1 - 1, x_2 y_2 - 1, \ldots, x_n y_n - 1).\]

AUTHORS:

  • David Roe (2008-2-23): created

  • David Loeffler (2009-07-10): cleaned up docstrings

sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing(base_ring, *args, **kwds)[source]

Return the globally unique univariate or multivariate Laurent polynomial ring with given properties and variable name or names.

There are four ways to call the Laurent polynomial ring constructor:

  1. LaurentPolynomialRing(base_ring, name,    sparse=False)

  2. LaurentPolynomialRing(base_ring, names,   order='degrevlex')

  3. LaurentPolynomialRing(base_ring, name, n, order='degrevlex')

  4. LaurentPolynomialRing(base_ring, n, name, order='degrevlex')

The optional arguments sparse and order must be explicitly named, and the other arguments must be given positionally.

INPUT:

  • base_ring – a commutative ring

  • name – string

  • names – list or tuple of names, or a comma separated string

  • n – positive integer

  • sparse – boolean (default: False); whether or not elements are sparse

  • order – string or TermOrder, e.g.,

    • 'degrevlex' – default; degree reverse lexicographic

    • 'lex' – lexicographic

    • 'deglex' – degree lexicographic

    • TermOrder('deglex',3) + TermOrder('deglex',3) – block ordering

OUTPUT:

LaurentPolynomialRing(base_ring, name, sparse=False) returns a univariate Laurent polynomial ring; all other input formats return a multivariate Laurent polynomial ring.

UNIQUENESS and IMMUTABILITY: In Sage there is exactly one single-variate Laurent polynomial ring over each base ring in each choice of variable and sparseness. There is also exactly one multivariate Laurent polynomial ring over each base ring for each choice of names of variables and term order.

sage: R.<x,y> = LaurentPolynomialRing(QQ, 2); R                                 # needs sage.modules
Multivariate Laurent Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^-2                                                          # needs sage.modules

>>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2); R                                 # needs sage.modules
Multivariate Laurent Polynomial Ring in x, y over Rational Field
>>> f = x**Integer(2) - Integer(2)*y**-Integer(2)                                                          # needs sage.modules

R.<x,y> = LaurentPolynomialRing(QQ, 2); R                                 # needs sage.modules
f = x^2 - 2*y^-2                                                          # needs sage.modules

You can’t just globally change the names of those variables. This is because objects all over Sage could have pointers to that polynomial ring.

sage: R._assign_names(['z','w'])                                                # needs sage.modules
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.

>>> from sage.all import *
>>> R._assign_names(['z','w'])                                                # needs sage.modules
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.

R._assign_names(['z','w'])                                                # needs sage.modules

EXAMPLES:

  1. LaurentPolynomialRing(base_ring, name, sparse=False)

    sage: LaurentPolynomialRing(QQ, 'w')
Univariate Laurent Polynomial Ring in w over Rational Field

    >>> from sage.all import *
>>> LaurentPolynomialRing(QQ, 'w')
Univariate Laurent Polynomial Ring in w over Rational Field

    LaurentPolynomialRing(QQ, 'w')

    Use the diamond brackets notation to make the variable ready for use after you define the ring:

    sage: R.<w> = LaurentPolynomialRing(QQ)
sage: (1 + w)^3
1 + 3*w + 3*w^2 + w^3

    >>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, names=('w',)); (w,) = R._first_ngens(1)
>>> (Integer(1) + w)**Integer(3)
1 + 3*w + 3*w^2 + w^3

    R.<w> = LaurentPolynomialRing(QQ)
(1 + w)^3

    You must specify a name:

    sage: LaurentPolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: you must specify the names of the variables

sage: R.<abc> = LaurentPolynomialRing(QQ, sparse=True); R
Univariate Laurent Polynomial Ring in abc over Rational Field

sage: R.<w> = LaurentPolynomialRing(PolynomialRing(GF(7),'k')); R
Univariate Laurent Polynomial Ring in w over
 Univariate Polynomial Ring in k over Finite Field of size 7

    >>> from sage.all import *
>>> LaurentPolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: you must specify the names of the variables

>>> R = LaurentPolynomialRing(QQ, sparse=True, names=('abc',)); (abc,) = R._first_ngens(1); R
Univariate Laurent Polynomial Ring in abc over Rational Field

>>> R = LaurentPolynomialRing(PolynomialRing(GF(Integer(7)),'k'), names=('w',)); (w,) = R._first_ngens(1); R
Univariate Laurent Polynomial Ring in w over
 Univariate Polynomial Ring in k over Finite Field of size 7

    LaurentPolynomialRing(QQ)
R.<abc> = LaurentPolynomialRing(QQ, sparse=True); R
R.<w> = LaurentPolynomialRing(PolynomialRing(GF(7),'k')); R

    Rings with different variables are different:

    sage: LaurentPolynomialRing(QQ, 'x') == LaurentPolynomialRing(QQ, 'y')
False

    >>> from sage.all import *
>>> LaurentPolynomialRing(QQ, 'x') == LaurentPolynomialRing(QQ, 'y')
False

    LaurentPolynomialRing(QQ, 'x') == LaurentPolynomialRing(QQ, 'y')

  2. LaurentPolynomialRing(base_ring, names,   order='degrevlex')

    sage: R = LaurentPolynomialRing(QQ, 'a,b,c'); R                              # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

sage: S = LaurentPolynomialRing(QQ, ['a','b','c']); S                        # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

sage: T = LaurentPolynomialRing(QQ, ('a','b','c')); T                        # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

    >>> from sage.all import *
>>> R = LaurentPolynomialRing(QQ, 'a,b,c'); R                              # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

>>> S = LaurentPolynomialRing(QQ, ['a','b','c']); S                        # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

>>> T = LaurentPolynomialRing(QQ, ('a','b','c')); T                        # needs sage.modules
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field

    R = LaurentPolynomialRing(QQ, 'a,b,c'); R                              # needs sage.modules
S = LaurentPolynomialRing(QQ, ['a','b','c']); S                        # needs sage.modules
T = LaurentPolynomialRing(QQ, ('a','b','c')); T                        # needs sage.modules

    All three rings are identical.

    sage: (R is S) and  (S is T)                                                 # needs sage.modules
True

    >>> from sage.all import *
>>> (R is S) and  (S is T)                                                 # needs sage.modules
True

    (R is S) and  (S is T)                                                 # needs sage.modules

    There is a unique Laurent polynomial ring with each term order:

    sage: # needs sage.modules
sage: R = LaurentPolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S = LaurentPolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S is LaurentPolynomialRing(QQ, 'x,y,z', order='invlex')
True
sage: R == S
False

    >>> from sage.all import *
>>> # needs sage.modules
>>> R = LaurentPolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
>>> S = LaurentPolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
>>> S is LaurentPolynomialRing(QQ, 'x,y,z', order='invlex')
True
>>> R == S
False

    # needs sage.modules
R = LaurentPolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
S = LaurentPolynomialRing(QQ, 'x,y,z', order='invlex'); S
S is LaurentPolynomialRing(QQ, 'x,y,z', order='invlex')
R == S

  3. LaurentPolynomialRing(base_ring, name, n, order='degrevlex')

    If you specify a single name as a string and a number of variables, then variables labeled with numbers are created.

    sage: LaurentPolynomialRing(QQ, 'x', 10)                                     # needs sage.modules
Multivariate Laurent Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
 over Rational Field

sage: LaurentPolynomialRing(GF(7), 'y', 5)                                   # needs sage.modules
Multivariate Laurent Polynomial Ring in y0, y1, y2, y3, y4
 over Finite Field of size 7

sage: LaurentPolynomialRing(QQ, 'y', 3, sparse=True)                         # needs sage.modules
Multivariate Laurent Polynomial Ring in y0, y1, y2 over Rational Field

    >>> from sage.all import *
>>> LaurentPolynomialRing(QQ, 'x', Integer(10))                                     # needs sage.modules
Multivariate Laurent Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
 over Rational Field

>>> LaurentPolynomialRing(GF(Integer(7)), 'y', Integer(5))                                   # needs sage.modules
Multivariate Laurent Polynomial Ring in y0, y1, y2, y3, y4
 over Finite Field of size 7

>>> LaurentPolynomialRing(QQ, 'y', Integer(3), sparse=True)                         # needs sage.modules
Multivariate Laurent Polynomial Ring in y0, y1, y2 over Rational Field

    LaurentPolynomialRing(QQ, 'x', 10)                                     # needs sage.modules
LaurentPolynomialRing(GF(7), 'y', 5)                                   # needs sage.modules
LaurentPolynomialRing(QQ, 'y', 3, sparse=True)                         # needs sage.modules

    By calling the inject_variables() method, all those variable names are available for interactive use:

    sage: R = LaurentPolynomialRing(GF(7), 15, 'w'); R                           # needs sage.modules
Multivariate Laurent Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7,
 w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7
sage: R.inject_variables()                                                   # needs sage.modules
Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14
sage: (w0 + 2*w8 + w13)^2                                                    # needs sage.modules
w0^2 + 4*w0*w8 + 4*w8^2 + 2*w0*w13 + 4*w8*w13 + w13^2

    >>> from sage.all import *
>>> R = LaurentPolynomialRing(GF(Integer(7)), Integer(15), 'w'); R                           # needs sage.modules
Multivariate Laurent Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7,
 w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7
>>> R.inject_variables()                                                   # needs sage.modules
Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14
>>> (w0 + Integer(2)*w8 + w13)**Integer(2)                                                    # needs sage.modules
w0^2 + 4*w0*w8 + 4*w8^2 + 2*w0*w13 + 4*w8*w13 + w13^2

    R = LaurentPolynomialRing(GF(7), 15, 'w'); R                           # needs sage.modules
R.inject_variables()                                                   # needs sage.modules
(w0 + 2*w8 + w13)^2                                                    # needs sage.modules
class sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_mpair(R)[source]

Bases: LaurentPolynomialRing_generic

EXAMPLES:

sage: L = LaurentPolynomialRing(QQ,2,'x')                                   # needs sage.modules
sage: type(L)                                                               # needs sage.modules
<class
'sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_mpair_with_category'>
sage: L == loads(dumps(L))                                                  # needs sage.modules
True

>>> from sage.all import *
>>> L = LaurentPolynomialRing(QQ,Integer(2),'x')                                   # needs sage.modules
>>> type(L)                                                               # needs sage.modules
<class
'sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_mpair_with_category'>
>>> L == loads(dumps(L))                                                  # needs sage.modules
True

L = LaurentPolynomialRing(QQ,2,'x')                                   # needs sage.modules
type(L)                                                               # needs sage.modules
L == loads(dumps(L))                                                  # needs sage.modules
Element[source]

alias of LaurentPolynomial_mpair

monomial(*exponents)[source]

Return the monomial whose exponents are given in argument.

EXAMPLES:

sage: # needs sage.modules
sage: L = LaurentPolynomialRing(QQ, 'x', 2)
sage: L.monomial(-3, 5)
x0^-3*x1^5
sage: L.monomial(1, 1)
x0*x1
sage: L.monomial(0, 0)
1
sage: L.monomial(-2, -3)
x0^-2*x1^-3

sage: x0, x1 = L.gens()                                                     # needs sage.modules
sage: L.monomial(-1, 2) == x0^-1 * x1^2                                     # needs sage.modules
True

sage: L.monomial(1, 2, 3)                                                   # needs sage.modules
Traceback (most recent call last):
...
TypeError: tuple key (1, 2, 3) must have same length as ngens (= 2)

>>> from sage.all import *
>>> # needs sage.modules
>>> L = LaurentPolynomialRing(QQ, 'x', Integer(2))
>>> L.monomial(-Integer(3), Integer(5))
x0^-3*x1^5
>>> L.monomial(Integer(1), Integer(1))
x0*x1
>>> L.monomial(Integer(0), Integer(0))
1
>>> L.monomial(-Integer(2), -Integer(3))
x0^-2*x1^-3

>>> x0, x1 = L.gens()                                                     # needs sage.modules
>>> L.monomial(-Integer(1), Integer(2)) == x0**-Integer(1) * x1**Integer(2)                                     # needs sage.modules
True

>>> L.monomial(Integer(1), Integer(2), Integer(3))                                                   # needs sage.modules
Traceback (most recent call last):
...
TypeError: tuple key (1, 2, 3) must have same length as ngens (= 2)

# needs sage.modules
L = LaurentPolynomialRing(QQ, 'x', 2)
L.monomial(-3, 5)
L.monomial(1, 1)
L.monomial(0, 0)
L.monomial(-2, -3)
x0, x1 = L.gens()                                                     # needs sage.modules
L.monomial(-1, 2) == x0^-1 * x1^2                                     # needs sage.modules
L.monomial(1, 2, 3)                                                   # needs sage.modules

We also allow to specify the exponents in a single tuple:

sage: L.monomial((-1, 2))                                                   # needs sage.modules
x0^-1*x1^2

sage: L.monomial((-1, 2, 3))                                                # needs sage.modules
Traceback (most recent call last):
...
TypeError: tuple key (-1, 2, 3) must have same length as ngens (= 2)

>>> from sage.all import *
>>> L.monomial((-Integer(1), Integer(2)))                                                   # needs sage.modules
x0^-1*x1^2

>>> L.monomial((-Integer(1), Integer(2), Integer(3)))                                                # needs sage.modules
Traceback (most recent call last):
...
TypeError: tuple key (-1, 2, 3) must have same length as ngens (= 2)

L.monomial((-1, 2))                                                   # needs sage.modules
L.monomial((-1, 2, 3))                                                # needs sage.modules
class sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_univariate(R)[source]

Bases: LaurentPolynomialRing_generic

EXAMPLES:

sage: L = LaurentPolynomialRing(QQ,'x')
sage: type(L)
<class 'sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_univariate_with_category'>
sage: TestSuite(L).run()

>>> from sage.all import *
>>> L = LaurentPolynomialRing(QQ,'x')
>>> type(L)
<class 'sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_univariate_with_category'>
>>> TestSuite(L).run()

L = LaurentPolynomialRing(QQ,'x')
type(L)
TestSuite(L).run()
Element[source]

alias of LaurentPolynomial_univariate

monomial(arg)[source]

Return the monomial with the given exponent.

sage.rings.polynomial.laurent_polynomial_ring.from_fraction_field(L, x)[source]

Helper function to construct a Laurent polynomial from an element of its parent’s fraction field.

INPUT:

OUTPUT:

An instance of the element class of L. If the denominator fails to be a unit in L an error is raised.

EXAMPLES:

sage: # needs sage.modules
sage: from sage.rings.polynomial.laurent_polynomial_ring import from_fraction_field
sage: L.<x, y> = LaurentPolynomialRing(ZZ)
sage: F = L.fraction_field()
sage: xi = F(~x)
sage: from_fraction_field(L, xi) == ~x
True

>>> from sage.all import *
>>> # needs sage.modules
>>> from sage.rings.polynomial.laurent_polynomial_ring import from_fraction_field
>>> L = LaurentPolynomialRing(ZZ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> F = L.fraction_field()
>>> xi = F(~x)
>>> from_fraction_field(L, xi) == ~x
True

# needs sage.modules
from sage.rings.polynomial.laurent_polynomial_ring import from_fraction_field
L.<x, y> = LaurentPolynomialRing(ZZ)
F = L.fraction_field()
xi = F(~x)
from_fraction_field(L, xi) == ~x
sage.rings.polynomial.laurent_polynomial_ring.is_LaurentPolynomialRing(R)[source]

Return True if and only if R is a Laurent polynomial ring.

EXAMPLES:

sage: from sage.rings.polynomial.laurent_polynomial_ring import is_LaurentPolynomialRing
sage: P = PolynomialRing(QQ, 2, 'x')
sage: is_LaurentPolynomialRing(P)
doctest:warning...
DeprecationWarning: is_LaurentPolynomialRing is deprecated; use isinstance(...,
sage.rings.polynomial.laurent_polynomial_ring_base.LaurentPolynomialRing_generic) instead
See https://github.com/sagemath/sage/issues/35229 for details.
False

sage: R = LaurentPolynomialRing(QQ,3,'x')                                       # needs sage.modules
sage: is_LaurentPolynomialRing(R)                                               # needs sage.modules
True

>>> from sage.all import *
>>> from sage.rings.polynomial.laurent_polynomial_ring import is_LaurentPolynomialRing
>>> P = PolynomialRing(QQ, Integer(2), 'x')
>>> is_LaurentPolynomialRing(P)
doctest:warning...
DeprecationWarning: is_LaurentPolynomialRing is deprecated; use isinstance(...,
sage.rings.polynomial.laurent_polynomial_ring_base.LaurentPolynomialRing_generic) instead
See https://github.com/sagemath/sage/issues/35229 for details.
False

>>> R = LaurentPolynomialRing(QQ,Integer(3),'x')                                       # needs sage.modules
>>> is_LaurentPolynomialRing(R)                                               # needs sage.modules
True

from sage.rings.polynomial.laurent_polynomial_ring import is_LaurentPolynomialRing
P = PolynomialRing(QQ, 2, 'x')
is_LaurentPolynomialRing(P)
R = LaurentPolynomialRing(QQ,3,'x')                                       # needs sage.modules
is_LaurentPolynomialRing(R)                                               # needs sage.modules