Elements of Laurent polynomial rings

Elements of Laurent polynomial rings

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair

Bases: sage.structure.element.CommutativeAlgebraElement

Currently, one can only create LaurentPolynomials out of dictionaries and elements of the base ring.

EXAMPLES:

sage: L.<w,z> = LaurentPolynomialRing(QQ)
sage: f = L({(-1,-1):1}); f
w^-1*z^-1
sage: f = L({(1,1):1}); f
w*z
sage: f =  L({(-1,-1):1, (1,3):4}); f
4*w*z^3 + w^-1*z^-1
sage: L(1/2)
1/2
coefficient(mon)

Return the coefficient of mon in self, where mon must have the same parent as self.

The coefficient is defined as follows. If \(f\) is this polynomial, then the coefficient \(c_m\) is sum:

\[c_m := \sum_T \frac{T}{m}\]

where the sum is over terms \(T\) in \(f\) that are exactly divisible by \(m\).

A monomial \(m(x,y)\) ‘exactly divides’ \(f(x,y)\) if \(m(x,y) | f(x,y)\) and neither \(x \cdot m(x,y)\) nor \(y \cdot m(x,y)\) divides \(f(x,y)\).

INPUT:

  • mon – a monomial

OUTPUT:

Element of the parent of self.

Note

To get the constant coefficient, call constant_coefficient().

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)

The coefficient returned is an element of the parent of self; in this case, P.

sage: f = 2 * x * y
sage: c = f.coefficient(x*y); c
2
sage: c.parent()
Multivariate Laurent Polynomial Ring in x, y over Rational Field

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^2 + 5*x*y)*x^-3; f
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2
sage: f.coefficient(y)
5*x^-2
sage: f.coefficient(y^2)
-7*x^-2 + x^-3
sage: f.coefficient(x*y)
0
sage: f.coefficient(x^-2)
-7*y^2 + 5*y
sage: f.coefficient(x^-2*y^2)
-7
sage: f.coefficient(1)
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2
coefficients()

Return the nonzero coefficients of this polynomial in a list. The returned list is decreasingly ordered by the term ordering of self.parent().

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ,order='degrevlex')
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.coefficients()
[4, 3, 2, 1]
sage: L.<x,y,z> = LaurentPolynomialRing(QQ,order='lex')
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.coefficients()
[4, 1, 2, 3]
constant_coefficient()

Return the constant coefficient of self.

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^2 + 5*x*y)*x^-3; f
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2
sage: f.constant_coefficient()
0
sage: f = (x^3 + 2*x^-2*y+y^3)*y^-3; f
x^3*y^-3 + 1 + 2*x^-2*y^-2
sage: f.constant_coefficient()
1
degree(x=None)

Returns the degree of x in self

EXAMPLES:

sage: R.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.degree(x)
7
sage: f.degree(y)
1
sage: f.degree(z)
0
dict()

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: list(sorted(f.dict().iteritems()))
[((3, 1, 0), 3), ((4, 0, -2), 2), ((6, -7, 0), 1), ((7, 0, -1), 4)]
exponents()

Returns a list of the exponents of self.

EXAMPLES:

sage: L.<w,z> = LaurentPolynomialRing(QQ)
sage: a = w^2*z^-1+3; a
w^2*z^-1 + 3
sage: e = a.exponents()
sage: e.sort(); e
[(0, 0), (2, -1)]
factor()

Returns a Laurent monomial (the unit part of the factorization) and a factored multi-polynomial.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.factor()
(x^3*y^-7*z^-2) * (4*x^4*y^7*z + 3*y^8*z^2 + 2*x*y^7 + x^3*z^2)
has_any_inverse()

Returns True if self contains any monomials with a negative exponent, False otherwise.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.has_any_inverse()
True
sage: g = x^2 + y^2
sage: g.has_any_inverse()
False
has_inverse_of(i)

INPUT:

  • i – The index of a generator of self.parent()

OUTPUT:

Returns True if self contains a monomial including the inverse of self.parent().gen(i), False otherwise.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.has_inverse_of(0)
False
sage: f.has_inverse_of(1)
True
sage: f.has_inverse_of(2)
True
monomial_coefficient(mon)

Return the coefficient in the base ring of the monomial mon in self, where mon must have the same parent as self.

This function contrasts with the function coefficient() which returns the coefficient of a monomial viewing this polynomial in a polynomial ring over a base ring having fewer variables.

INPUT:

  • mon - a monomial

See also

For coefficients in a base ring of fewer variables, see coefficient().

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: f.monomial_coefficient(x^-2*y^3)
-7
sage: f.monomial_coefficient(x^2)
0
monomials()

Return the list of monomials in self.

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: f.monomials()
[x^6, x^-3*y^2, x^-2*y, x^-2*y^3]
subs(in_dict=None, **kwds)

Note that this is a very unsophisticated implementation.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = x + 2*y + 3*z
sage: f.subs(x=1)
2*y + 3*z + 1
sage: f.subs(y=1)
x + 3*z + 2
sage: f.subs(z=1)
x + 2*y + 3
sage: f.subs(x=1,y=1,z=1)
6

sage: f = x^-1
sage: f.subs(x=2)
1/2
sage: f.subs({x:2})
1/2

sage: f = x + 2*y + 3*z
sage: f.subs({x:1,y:1,z:1})
6
sage: f.substitute(x=1,y=1,z=1)
6

TESTS:

sage: f = x + 2*y + 3*z
sage: f(q=10)
x + 2*y + 3*z
variables(sort=True)

Return a tuple of all variables occurring in self.

INPUT:

  • sort – specifies whether the indices shall be sorted

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.variables()
(z, y, x)
sage: f.variables(sort=False) #random
(y, z, x)

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