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)

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

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

Python

>>> 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

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

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

Python

>>> 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.

EXAMPLES:

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

   Sage

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

   Python

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

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

   Sage

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

   Python

   >>> 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
   

   You must specify a name:

   Sage

   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
   

   Python

   >>> 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
   

   Rings with different variables are different:

   Sage

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

   Python

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

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

   Sage

   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
   

   Python

   >>> 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
   

   All three rings are identical.

   Sage

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

   Python

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

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

   Sage

   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
   

   Python

   >>> 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
   

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

   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
   

   Python

   >>> 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
   

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

   Sage

   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
   

   Python

   >>> 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
   

class sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_mpair(R)

Bases: LaurentPolynomialRing_generic

EXAMPLES:

Sage

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

Python

>>> 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

Element

alias of LaurentPolynomial_mpair

monomial(*exponents)

Return the monomial whose exponents are given in argument.

EXAMPLES:

Sage

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)

Python

>>> 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)

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

Sage

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)

Python

>>> 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)

class sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_univariate(R)

Bases: LaurentPolynomialRing_generic

EXAMPLES:

Sage

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

Python

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

Element

alias of LaurentPolynomial_univariate

monomial(arg)

Return the monomial with the given exponent.

sage.rings.polynomial.laurent_polynomial_ring.from_fraction_field(L, x)

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

INPUT:

• L – an instance of LaurentPolynomialRing_generic

• x – an element of the fraction field of L

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

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

Python

>>> 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

sage.rings.polynomial.laurent_polynomial_ring.is_LaurentPolynomialRing(R)

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

EXAMPLES:

Sage

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

Python

>>> 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