Ring of Laurent Polynomials (base class)

If $R$ is a commutative ring, then the ring of Laurent polynomials in $n$ variables over $R$ is $R[x_1^{\pm 1},x_2^{\pm 1},\dots ,x_n^{\pm 1}]$.

AUTHORS:

• David Roe (2008-2-23): created

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

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

Bases: Parent

Laurent polynomial ring (base class).

EXAMPLES:

Since Issue #11900, it is in the category of commutative rings:

sage: R.<x1,x2> = LaurentPolynomialRing(QQ)
sage: R.category()
Join of Category of unique factorization domains
    and Category of algebras with basis
        over (number fields and quotient fields and metric spaces)
    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)                                                        # needs sage.rings.number_field
sage: P.change_ring(K)                                                                  # needs sage.rings.number_field
Multivariate Laurent Polynomial Ring in x over
 Cyclotomic Field of order 4 and degree 2

characteristic()

Return the characteristic of the base ring.

EXAMPLES:

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

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

Return the completion of self.

Currently only implemented for the ring of formal Laurent series. The prec variable controls the precision used in the Laurent series ring. If prec is $\mathrm{\infty }$, then this returns a LazyLaurentSeriesRing.

EXAMPLES:

sage: P.<x> = LaurentPolynomialRing(QQ); P
Univariate Laurent Polynomial Ring in x over Rational Field
sage: PP = P.completion(x); PP
Laurent Series Ring in x over Rational Field
sage: f = 1 - 1/x
sage: PP(f)
-x^-1 + 1
sage: g = 1 / PP(f); g
-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)
sage: 1 / g
-x^-1 + 1 + O(x^19)

sage: # needs sage.combinat
sage: PP = P.completion(x, prec=oo); PP
Lazy Laurent Series Ring in x over Rational Field
sage: g = 1 / PP(f); g
-x - x^2 - x^3 + O(x^4)
sage: 1 / g == f
True

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)

Return 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

gens()

Return the tuple of generators of self.

EXAMPLES:

sage: LaurentPolynomialRing(ZZ, 2, 'x').gens()
(x0, x1)
sage: LaurentPolynomialRing(QQ, 1, 'x').gens()
(x,)

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()

Return 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)

Return True if self is an integral domain.

EXAMPLES:

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

The following used to fail; see Issue #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()

Return True if self is Noetherian.

EXAMPLES:

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

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()

Return 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(min_valuation=-2, max_degree=2, *args, **kwds)

Return a random polynomial with degree at most max_degree and lowest valuation at least min_valuation.

Uses the random sampling from the base polynomial ring then divides out by a monomial to ensure correct max_degree and min_valuation.

INPUT:

• min_valuation – integer (default: $-2$); the minimal allowed valuation of the polynomial

• max_degree – integer (default: $2$); the maximal allowed degree of the polynomial

• *args, **kwds – passed to the random element generator of the base polynomial ring and base ring itself

EXAMPLES:

sage: L.<x> = LaurentPolynomialRing(QQ)
sage: f = L.random_element()
sage: f.degree() <= 2
True
sage: f.valuation() >= -2
True
sage: f.parent() is L
True

sage: L = LaurentPolynomialRing(ZZ, 2, 'x')
sage: f = L.random_element(10, 20)
sage: f.degree() <= 20
True
sage: f.valuation() >= 10
True
sage: f.parent() is L
True

sage: L = LaurentPolynomialRing(GF(13), 3, 'x')
sage: f = L.random_element(-10, -1)
sage: f.degree() <= -1
True
sage: f.valuation() >= -10
True
sage: f.parent() is L
True

sage: L.<x, y> = LaurentPolynomialRing(RR)
sage: f = L.random_element()
sage: f.degree() <= 2
True
sage: f.valuation() >= -2
True
sage: f.parent() is L
True

sage: L = LaurentPolynomialRing(QQbar, 5, 'x')
sage: f = L.random_element(-1, 1)
sage: f = L.random_element(-1, 1)
sage: f.degree() <= 1
True
sage: f.valuation() >= -1
True
sage: f.parent() is L
True

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()

Return 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 – 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')