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 – a string

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

• n – a positive integer

• sparse – bool (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
Multivariate Laurent Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^-2

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'])
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.

EXAMPLES:

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

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

   Rings with different variables are different:

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

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

   sage: R = LaurentPolynomialRing(QQ, 'a,b,c'); R
   Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
   
   sage: S = LaurentPolynomialRing(QQ, ['a','b','c']); S
   Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
   
   sage: T = LaurentPolynomialRing(QQ, ('a','b','c')); T
   Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
   

   All three rings are identical.

   sage: (R is S) and  (S is T)
   True
   

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

   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
   

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)
   Multivariate Laurent Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field
   
   sage: LaurentPolynomialRing(GF(7), 'y', 5)
   Multivariate Laurent Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7
   
   sage: LaurentPolynomialRing(QQ, 'y', 3, sparse=True)
   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: R = LaurentPolynomialRing(GF(7),15,'w'); R
   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()
   Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14
   sage: (w0 + 2*w8 + w13)^2
   w0^2 + 4*w0*w8 + 4*w8^2 + 2*w0*w13 + 4*w8*w13 + w13^2
   

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

Bases: CommutativeRing, Parent

Laurent polynomial ring (base class).

EXAMPLES:

This base class inherits from CommutativeRing. Since trac ticket #11900, it is also initialised as such:

sage: R.<x1,x2> = LaurentPolynomialRing(QQ)
sage: R.category()
Join of Category of unique factorization domains and Category of commutative algebras over (number fields and quotient fields and metric spaces) and Category of infinite sets
sage: TestSuite(R).run()

change_ring(base_ring=None, names=None, sparse=False, order=None)

EXAMPLES:

sage: R = LaurentPolynomialRing(QQ,2,'x')
sage: R.change_ring(ZZ)
Multivariate Laurent Polynomial Ring in x0, x1 over Integer Ring

Check that the distinction between a univariate ring and a multivariate ring with one generator is preserved:

sage: P.<x> = LaurentPolynomialRing(QQ, 1)
sage: P
Multivariate Laurent Polynomial Ring in x over Rational Field
sage: K.<i> = CyclotomicField(4)
sage: P.change_ring(K)
Multivariate Laurent Polynomial Ring in x over Cyclotomic Field of order 4 and degree 2

characteristic()

Returns the characteristic of the base ring.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').characteristic()
0
sage: LaurentPolynomialRing(GF(3),2,'x').characteristic()
3

completion(p, prec=20, extras=None)

EXAMPLES:

sage: P.<x>=LaurentPolynomialRing(QQ)
sage: P
Univariate Laurent Polynomial Ring in x over Rational Field
sage: PP=P.completion(x)
sage: PP
Laurent Series Ring in x over Rational Field
sage: f=1-1/x
sage: PP(f)
-x^-1 + 1
sage: 1/PP(f)
-x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14 - x^15 - x^16 - x^17 - x^18 - x^19 - x^20 + O(x^21)

construction()

Return the construction of self.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x,y').construction()
(LaurentPolynomialFunctor,
Univariate Laurent Polynomial Ring in x over Rational Field)

fraction_field()

The fraction field is the same as the fraction field of the polynomial ring.

EXAMPLES:

sage: L.<x> = LaurentPolynomialRing(QQ)
sage: L.fraction_field()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: (x^-1 + 2) / (x - 1)
(2*x + 1)/(x^2 - x)

gen(i=0)

Returns the \(i^{th}\) generator of self. If i is not specified, then the first generator will be returned.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').gen()
x0
sage: LaurentPolynomialRing(QQ,2,'x').gen(0)
x0
sage: LaurentPolynomialRing(QQ,2,'x').gen(1)
x1

ideal(*args, **kwds)

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').ideal([1])
Ideal (1) of Multivariate Laurent Polynomial Ring in x0, x1 over Rational Field

is_exact()

Returns True if the base ring is exact.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').is_exact()
True
sage: LaurentPolynomialRing(RDF,2,'x').is_exact()
False

is_field(proof=True)

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').is_field()
False

is_finite()

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').is_finite()
False

is_integral_domain(proof=True)

Returns True if self is an integral domain.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').is_integral_domain()
True

The following used to fail; see trac ticket #7530:

sage: L = LaurentPolynomialRing(ZZ, 'X')
sage: L['Y']
Univariate Polynomial Ring in Y over Univariate Laurent Polynomial Ring in X over Integer Ring

is_noetherian()

Returns True if self is Noetherian.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').is_noetherian()
Traceback (most recent call last):
...
NotImplementedError

krull_dimension()

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').krull_dimension()
Traceback (most recent call last):
...
NotImplementedError

ngens()

Return the number of generators of self.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').ngens()
2
sage: LaurentPolynomialRing(QQ,1,'x').ngens()
1

polynomial_ring()

Returns the polynomial ring associated with self.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').polynomial_ring()
Multivariate Polynomial Ring in x0, x1 over Rational Field
sage: LaurentPolynomialRing(QQ,1,'x').polynomial_ring()
Multivariate Polynomial Ring in x over Rational Field

random_element(low_degree=-2, high_degree=2, terms=5, choose_degree=False, *args, **kwds)

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').random_element()
Traceback (most recent call last):
...
NotImplementedError

remove_var(var)

EXAMPLES:

sage: R = LaurentPolynomialRing(QQ,'x,y,z')
sage: R.remove_var('x')
Multivariate Laurent Polynomial Ring in y, z over Rational Field
sage: R.remove_var('x').remove_var('y')
Univariate Laurent Polynomial Ring in z over Rational Field

term_order()

Returns the term order of self.

EXAMPLES:

sage: LaurentPolynomialRing(QQ,2,'x').term_order()
Degree reverse lexicographic term order

variable_names_recursive(depth=+Infinity)

Return the list of variable names of this ring and its base rings, as if it were a single multi-variate Laurent polynomial.

INPUT:

• depth – an integer or Infinity.

OUTPUT:

A tuple of strings.

EXAMPLES:

sage: T = LaurentPolynomialRing(QQ, 'x')
sage: S = LaurentPolynomialRing(T, 'y')
sage: R = LaurentPolynomialRing(S, 'z')
sage: R.variable_names_recursive()
('x', 'y', 'z')
sage: R.variable_names_recursive(2)
('y', 'z')

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

Bases: LaurentPolynomialRing_generic

EXAMPLES:

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

Element

alias of LaurentPolynomial_mpair

monomial(*args)

Return the monomial whose exponents are given in argument.

EXAMPLES:

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

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

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

Bases: LaurentPolynomialRing_generic

EXAMPLES:

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

Element

alias of LaurentPolynomial_univariate

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: 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

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

Returns 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)
False

sage: R = LaurentPolynomialRing(QQ,3,'x')
sage: is_LaurentPolynomialRing(R)
True