Elements of Laurent polynomial rings
Bases: sage.structure.element.CommutativeAlgebraElement
A generic Laurent polynomial.
Bases: sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_generic
Multivariate Laurent polynomials.
Return the coefficient of mon in self, where mon must have the same parent as self.
The coefficient is defined as follows. If is this polynomial, then
the coefficient
is sum:
where the sum is over terms in
that are exactly divisible
by
.
A monomial ‘exactly divides’
if
and neither
nor
divides
.
INPUT:
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
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]
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
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
The formal derivative of this Laurent polynomial, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.
See also
_derivative()
EXAMPLES:
sage: R = LaurentPolynomialRing(ZZ,'x, y')
sage: x, y = R.gens()
sage: t = x**4*y+x*y+y+x**(-1)+y**(-3)
sage: t.derivative(x, x)
12*x^2*y + 2*x^-3
sage: t.derivative(y, 2)
12*y^-5
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)]
The formal derivative of this Laurent polynomial, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.
See also
_derivative()
EXAMPLES:
sage: R = LaurentPolynomialRing(ZZ,'x, y')
sage: x, y = R.gens()
sage: t = x**4*y+x*y+y+x**(-1)+y**(-3)
sage: t.derivative(x, x)
12*x^2*y + 2*x^-3
sage: t.derivative(y, 2)
12*y^-5
The formal derivative of this Laurent polynomial, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.
See also
_derivative()
EXAMPLES:
sage: R = LaurentPolynomialRing(ZZ,'x, y')
sage: x, y = R.gens()
sage: t = x**4*y+x*y+y+x**(-1)+y**(-3)
sage: t.derivative(x, x)
12*x^2*y + 2*x^-3
sage: t.derivative(y, 2)
12*y^-5
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)]
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)
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
INPUT:
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
Return True if this element is a monomial.
EXAMPLES:
sage: k.<y,z> = LaurentPolynomialRing(QQ)
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^-2909).is_monomial()
True
sage: (38*z^-2909).is_monomial()
False
Return True if this is a univariate or constant Laurent polynomial, and False otherwise.
EXAMPLES:
sage: R.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = (x^3 + y^-3)*z
sage: f.is_univariate()
False
sage: g = f(1,y,4)
sage: g.is_univariate()
True
sage: R(1).is_univariate()
True
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:
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
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]
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
Returns a univariate polynomial associated to this multivariate polynomial.
INPUT:
If this polynomial is not in at most one variable, then a ValueError exception is raised. The new polynomial is over the same base ring as the given LaurentPolynomial and in the variable x if no ring R is provided.
EXAMPLES:
sage: R.<x, y> = LaurentPolynomialRing(ZZ)
sage: f = 3*x^2 - 2*y^-1 + 7*x^2*y^2 + 5
sage: f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
sage: g = f(10,y); g
700*y^2 + 305 - 2*y^-1
sage: h = g.univariate_polynomial(); h
-2*y^-1 + 305 + 700*y^2
sage: h.parent()
Univariate Laurent Polynomial Ring in y over Integer Ring
sage: g.univariate_polynomial(LaurentPolynomialRing(QQ,'z'))
-2*z^-1 + 305 + 700*z^2
Here’s an example with a constant multivariate polynomial:
sage: g = R(1)
sage: h = g.univariate_polynomial(); h
1
sage: h.parent()
Univariate Laurent Polynomial Ring in x over Integer Ring
Return a tuple of all variables occurring in self.
INPUT:
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)
Bases: sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_generic
A univariate Laurent polynomial in the form of
where
is a polynomial in
.
INPUT:
AUTHORS:
Return a copy of this Laurent polynomial, with coefficients in R.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: a = x^2 + 3*x^3 + 5*x^-1
sage: a.change_ring(GF(3))
2*x^-1 + x^2
Return the nonzero coefficients of self.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.coefficients()
[-5, 1, 1, -10/3]
Return the coefficient of the constant term of self.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = 3*t^-2 - t^-1 + 3 + t^2
sage: f.constant_coefficient()
3
sage: g = -2*t^-2 + t^-1 + 3*t
sage: g.constant_coefficient()
0
Return the degree of this polynomial.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: g = x^2 - x^4
sage: g.degree()
4
sage: g = -10/x^5 + x^2 - x^7
sage: g.degree()
7
The formal derivative of this Laurent polynomial, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global :func`derivative()` function for more details.
See also
_derivative()
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: g = 1/x^10 - x + x^2 - x^4
sage: g.derivative()
-10*x^-11 - 1 + 2*x - 4*x^3
sage: g.derivative(x)
-10*x^-11 - 1 + 2*x - 4*x^3
sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = LaurentPolynomialRing(R)
sage: f = 2*t/x + (3*t^2 + 6*t)*x
sage: f.derivative()
-2*t*x^-2 + (3*t^2 + 6*t)
sage: f.derivative(x)
-2*t*x^-2 + (3*t^2 + 6*t)
sage: f.derivative(t)
2*x^-1 + (6*t + 6)*x
Return a dictionary representing self.
Return the exponents appearing in self with nonzero coefficients.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.exponents()
[-2, 1, 2, 3]
Return a Laurent monomial (the unit part of the factorization) and a factored polynomial.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(ZZ)
sage: f = 4*t^-7 + 3*t^3 + 2*t^4 + t^-6
sage: f.factor()
(t^-7) * (4 + t + 3*t^10 + 2*t^11)
Return the gcd of self with right where the common divisor d makes both self and right into polynomials with the lowest possible degree.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: t.gcd(2)
1
sage: gcd(t^-2 + 1, t^-4 + 3*t^-1)
t^-4
sage: gcd((t^-2 + t)*(t + t^-1), (t^5 + t^8)*(1 + t^-2))
t^-3 + t^-1 + 1 + t^2
The formal integral of this Laurent series with 0 constant term.
EXAMPLES:
The integral may or may not be defined if the base ring is not a field.
sage: t = LaurentPolynomialRing(ZZ, 't').0
sage: f = 2*t^-3 + 3*t^2
sage: f.integral()
-t^-2 + t^3
sage: f = t^3
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: coefficients of integral cannot be coerced into the base ring
The integral of is
, which is not given by a
Laurent polynomial:
sage: t = LaurentPolynomialRing(ZZ,'t').0
sage: f = -1/t^3 - 31/t
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: the integral of is not a Laurent polynomial, since t^-1 has nonzero coefficient
Another example with just one negative coefficient:
sage: A.<t> = LaurentPolynomialRing(QQ)
sage: f = -2*t^(-4)
sage: f.integral()
2/3*t^-3
sage: f.integral().derivative() == f
True
Return the inverse of self if a unit.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (t^-2).inverse_of_unit()
t^2
sage: (t + 2).inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: element is not a unit
Return True if self is constant.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: x.is_constant()
False
sage: R.one().is_constant()
True
sage: (x^-2).is_constant()
False
sage: (x^2).is_constant()
False
sage: (x^-2 + 2).is_constant()
False
Return True if this element is a monomial. That is, if self is
for some integer
.
EXAMPLES:
sage: k.<z> = LaurentPolynomialRing(QQ)
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^-2909).is_monomial()
True
sage: (38*z^-2909).is_monomial()
False
Return True if this Laurent polynomial is a unit in this ring.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (2+t).is_unit()
False
sage: f = 2*t
sage: f.is_unit()
True
sage: 1/f
1/2*t^-1
sage: R(0).is_unit()
False
sage: R.<s> = LaurentPolynomialRing(ZZ)
sage: g = 2*s
sage: g.is_unit()
False
sage: 1/g
1/2*s^-1
ALGORITHM: A Laurent polynomial is a unit if and only if its “unit part” is a unit.
Return 1 if self is 0, else return 0.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x + x^2 + 3*x^4
sage: f.is_zero()
0
sage: z = 0*f
sage: z.is_zero()
1
Return the polynomial and the shift in power used to construct the
Laurent polynomial .
OUTPUT:
A tuple (u, n) where u is the underlying polynomial and n is the power of the exponent shift.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.polynomial_construction()
(3*x^5 + x^3 + 1, -1)
Attempts to divide self by right and returns a quotient and a remainder.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: (t^-3 - t^3).quo_rem(t^-1 - t)
(t^-2 + 1 + t^2, 0)
sage: (t^-2 + 3 + t).quo_rem(t^-4)
(t^2 + 3*t^4 + t^5, 0)
sage: (t^-2 + 3 + t).quo_rem(t^-4 + t)
(0, 1 + 3*t^2 + t^3)
Return the residue of self.
The residue is the coefficient of .
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ)
sage: f = 3*t^-2 - t^-1 + 3 + t^2
sage: f.residue()
-1
sage: g = -2*t^-2 + 4 + 3*t
sage: g.residue()
0
sage: f.residue().parent()
Rational Field
Return this Laurent polynomial multiplied by the power .
Does not change this polynomial.
EXAMPLES:
sage: R.<t> = LaurentPolynomialRing(QQ['y'])
sage: f = (t+t^-1)^4; f
t^-4 + 4*t^-2 + 6 + 4*t^2 + t^4
sage: f.shift(10)
t^6 + 4*t^8 + 6*t^10 + 4*t^12 + t^14
sage: f >> 10
t^-14 + 4*t^-12 + 6*t^-10 + 4*t^-8 + t^-6
sage: f << 4
1 + 4*t^2 + 6*t^4 + 4*t^6 + t^8
Return a polynomial with degree at most whose
-th coefficients
agree with self for all
.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x^12 + x^3 + x^5 + x^9
sage: f.truncate(10)
x^-12 + x^3 + x^5 + x^9
sage: f.truncate(5)
x^-12 + x^3
sage: f.truncate(-16)
0
Return the valuation of self.
The valuation of a Laurent polynomial is
plus the
valuation of
.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = 1/x + x^2 + 3*x^4
sage: g = 1 - x + x^2 - x^4
sage: f.valuation()
-1
sage: g.valuation()
0
Return the name of variable of self as a string.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.variable_name()
'x'
Return the tuple of variables occuring in this Laurent polynomial.
EXAMPLES:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: f = 1/x + x^2 + 3*x^4
sage: f.variables()
(x,)
sage: R.one().variables()
()