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)

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()
Category of commutative rings
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
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()

EXAMPLES:

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

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)

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
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)

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
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