If is a commutative ring, then the ring of Laurent polynomials in variables over is . We implement it as a quotient ring
TESTS:
sage: P.<q> = LaurentPolynomialRing(QQ)
sage: qi = q^(-1)
sage: qi in P
True
sage: P(qi)
q^-1
sage: A.<Y> = QQ[]
sage: R.<X> = LaurentPolynomialRing(A)
sage: matrix(R,2,2,[X,0,0,1])
[X 0]
[0 1]
AUTHORS:
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:
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'])
...
ValueError: variable names cannot be changed after object creation.
EXAMPLES:
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
w^3 + 3*w^2 + 3*w + 1
You must specify a name:
sage: LaurentPolynomialRing(QQ)
...
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
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
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
Bases: sage.rings.ring.CommutativeRing, sage.structure.parent_gens.ParentWithGens
EXAMPLES:
sage: R = LaurentPolynomialRing(QQ,2,'x')
sage: R.change_ring(ZZ)
Multivariate Laurent Polynomial Ring in x0, x1 over Integer Ring
Returns the characteristic of the base ring.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').characteristic()
0
sage: LaurentPolynomialRing(GF(3),2,'x').characteristic()
3
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').completion(3)
...
NotImplementedError
Returns the construction of self.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x,y').construction()
(LaurentPolynomialFunctor,
Univariate Laurent Polynomial Ring in x over Rational Field)
Returns the 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
TESTS:
sage: LaurentPolynomialRing(QQ,2,'x').gen(3)
...
ValueError: generator not defined
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').ideal()
...
NotImplementedError
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
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').is_field()
False
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').is_finite()
False
Returns True if self is an integral domain.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').is_integral_domain()
True
The following used to fail; see #7530:
sage: L = LaurentPolynomialRing(ZZ, 'X')
sage: L['Y']
Univariate Polynomial Ring in Y over Univariate Laurent Polynomial Ring in X over Integer Ring
Returns True if self is Noetherian.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').is_noetherian()
...
NotImplementedError
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').krull_dimension()
...
NotImplementedError
Returns the number of generators of self.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').ngens()
2
sage: LaurentPolynomialRing(QQ,1,'x').ngens()
1
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
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').random_element()
...
NotImplementedError
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
Returns the term order of self.
EXAMPLES:
sage: LaurentPolynomialRing(QQ,2,'x').term_order()
Degree reverse lexicographic term order
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