Module laurent_polynomial_ring

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

EXAMPLES:
    sage: P = PolynomialRing(QQ,2,'x')
    sage: is_LaurentPolynomialRing(P)
    False

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

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 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
    w^3 + 3*w^2 + 3*w + 1
    
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

You can call \code{injvar}, which is a convenient shortcut for \code{inject_variables()}.
    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.injvar()
    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