Elements of Laurent polynomial rings¶

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial¶

Bases: sage.structure.element.CommutativeAlgebraElement

Base class for Laurent polynomials.

change_ring(R)¶

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

Check that trac ticket #22277 is fixed:

sage: R.<x, y> = LaurentPolynomialRing(QQ)
sage: a = 2*x^2 + 3*x^3 + 4*x^-1
sage: a.change_ring(GF(3))
-x^2 + x^-1

dict()¶

Abstract dict method.

EXAMPLES:

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
sage: LaurentPolynomial.dict(x)
Traceback (most recent call last):
...
NotImplementedError

hamming_weight()¶

Return the hamming weight of self.

The hamming weight is number of non-zero coefficients and also known as the weight or sparsity.

EXAMPLES:

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - 1
sage: f.hamming_weight()
2

map_coefficients(f, new_base_ring=None)¶

Apply f to the coefficients of self.

If f is a sage.categories.map.Map, then the resulting polynomial will be defined over the codomain of f. Otherwise, the resulting polynomial will be over the same ring as self. Set new_base_ring to override this behavior.

INPUT:

f – a callable that will be applied to the coefficients of self.
new_base_ring (optional) – if given, the resulting polynomial will be defined over this ring.

EXAMPLES:

sage: k.<a> = GF(9)
sage: R.<x> = LaurentPolynomialRing(k)
sage: f = x*a + a
sage: f.map_coefficients(lambda a : a + 1)
(a + 1) + (a + 1)*x
sage: R.<x,y> = LaurentPolynomialRing(k, 2)
sage: f = x*a + 2*x^3*y*a + a
sage: f.map_coefficients(lambda a : a + 1)
(2*a + 1)*x^3*y + (a + 1)*x + a + 1

Examples with different base ring:

sage: R.<r> = GF(9); S.<s> = GF(81)
sage: h = Hom(R,S)[0]; h
Ring morphism:
  From: Finite Field in r of size 3^2
  To:   Finite Field in s of size 3^4
  Defn: r |--> 2*s^3 + 2*s^2 + 1
sage: T.<X,Y> = LaurentPolynomialRing(R, 2)
sage: f = r*X+Y
sage: g = f.map_coefficients(h); g
(2*s^3 + 2*s^2 + 1)*X + Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y over Finite Field in s of size 3^4
sage: h = lambda x: x.trace()
sage: g = f.map_coefficients(h); g
X - Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y over Finite Field in r of size 3^2
sage: g = f.map_coefficients(h, new_base_ring=GF(3)); g
X - Y
sage: g.parent()
Multivariate Laurent Polynomial Ring in X, Y over Finite Field of size 3

number_of_terms()¶

Abstract method for number of terms

EXAMPLES:

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
sage: LaurentPolynomial.number_of_terms(x)
Traceback (most recent call last):
...
NotImplementedError

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair¶

Bases: sage.rings.polynomial.laurent_polynomial.LaurentPolynomial

Multivariate Laurent polynomials.

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 self 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)¶

Return 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

derivative(*args)¶

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

dict()¶

Return self represented as a 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: sorted(f.dict().items())
[((3, 1, 0), 3), ((4, 0, -2), 2), ((6, -7, 0), 1), ((7, 0, -1), 4)]

diff(*args)¶

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

differentiate(*args)¶

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

exponents()¶

Return 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

is_constant()¶

Return whether this Laurent polynomial is constant.

EXAMPLES:

sage: L.<a, b> = LaurentPolynomialRing(QQ)
sage: L(0).is_constant()
True
sage: L(42).is_constant()
True
sage: a.is_constant()
False
sage: (1/b).is_constant()
False

is_monomial()¶

Return True if self 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

is_square(root=False)¶

Test whether this Laurent polynomial is a square.

INPUT:

root - boolean (default False) - if set to True then return a pair (True, sqrt) with sqrt a square root of this Laurent polynomial when it exists or (False, None).

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: p = (1 + x*y + z^-3)
sage: (p**2).is_square()
True
sage: (p**2).is_square(root=True)
(True, x*y + 1 + z^-3)

sage: x.is_square()
False
sage: x.is_square(root=True)
(False, None)

sage: (x**-4 * (1 + z)).is_square(root=False)
False
sage: (x**-4 * (1 + z)).is_square(root=True)
(False, None)

is_unit()¶

Return True if self is a unit.

The ground ring is assumed to be an integral domain.

This means that the Laurent polynomial is a monomial with unit coefficient.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ)
sage: (x*y/2).is_unit()
True
sage: (x + y).is_unit()
False
sage: (L.zero()).is_unit()
False
sage: (L.one()).is_unit()
True

sage: L.<x,y> = LaurentPolynomialRing(ZZ)
sage: (2*x*y).is_unit()
False

is_univariate()¶

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

iterator_exp_coeff()¶

Iterate over self as pairs of (ETuple, coefficient).

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: list(f.iterator_exp_coeff())
[((6, 0), -1), ((-2, 3), -7), ((-2, 1), 5), ((-3, 2), 1)]

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: sorted(f.monomials())
[x^-3*y^2, x^-2*y, x^-2*y^3, x^6]

number_of_terms()¶

Return the number of non-zero coefficients of self.

Also called weight, hamming weight or sparsity.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - y
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+1/y)^100
sage: f.number_of_terms()
101

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

quo_rem(right)¶

Divide this Laurent polynomial by right and return a quotient and a remainder.

INPUT:

right – a Laurent polynomial

OUTPUT:

A pair of Laurent polynomials.

EXAMPLES:

sage: R.<s, t> = LaurentPolynomialRing(QQ)
sage: (s^2-t^2).quo_rem(s-t)
(s + t, 0)
sage: (s^-2-t^2).quo_rem(s-t)
(s + t, -s^2 + s^-2)
sage: (s^-2-t^2).quo_rem(s^-1-t)
(t + s^-1, 0)

rescale_vars(d, h=None, new_ring=None)¶

Rescale variables in a Laurent polynomial.

INPUT:

d – a dict whose keys are the generator indices and values are the coefficients; so a pair (i, v) means \(x_i \mapsto v x_i\)
h – (optional) a map to be applied to coefficients done after rescaling
new_ring – (optional) a new ring to map the result into

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = x^-2*y + x*y^-2
sage: p.rescale_vars({0: 2, 1: 3})
2/9*x*y^-2 + 3/4*x^-2*y
sage: F = GF(2)
sage: p.rescale_vars({0: 3, 1: 7}, new_ring=L.change_ring(F))
x*y^-2 + x^-2*y

Test for trac ticket #30331:

sage: F.<z> = CyclotomicField(3)
sage: p.rescale_vars({0: 2, 1: z}, new_ring=L.change_ring(F))
2*z*x*y^-2 + 1/4*z*x^-2*y

subs(in_dict=None, **kwds)¶

Substitute some variables in this Laurent polynomial.

Variable/value pairs for the substitution may be given as a dictionary or via keyword-value pairs. If both are present, the latter take precedence.

INPUT:

in_dict – dictionary (optional)
**kwargs – keyword arguments

OUTPUT:

A Laurent polynomial.

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

toric_coordinate_change(M, h=None, new_ring=None)¶

Apply a matrix to the exponents in a Laurent polynomial.

For efficiency, we implement this directly, rather than as a substitution.

The optional argument h is a map to be applied to coefficients.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = 2*x^2 + y - x*y
sage: p.toric_coordinate_change(Matrix([[1,-3],[1,1]]))
2*x^2*y^2 - x^-2*y^2 + x^-3*y
sage: F = GF(2)
sage: p.toric_coordinate_change(Matrix([[1,-3],[1,1]]), new_ring=L.change_ring(F))
x^-2*y^2 + x^-3*y

toric_substitute(v, v1, a, h=None, new_ring=None)¶

Perform a single-variable substitution up to a toric coordinate change.

The optional argument h is a map to be applied to coefficients.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = x + y
sage: p.toric_substitute((2,3), (-1,1), 2)
1/2*x^3*y^3 + 2*x^-2*y^-2
sage: F = GF(5)
sage: p.toric_substitute((2,3), (-1,1), 2, new_ring=L.change_ring(F))
3*x^3*y^3 + 2*x^-2*y^-2

univariate_polynomial(R=None)¶

Returns a univariate polynomial associated to this multivariate polynomial.

INPUT:

R - (default: None) a univariate Laurent polynomial ring

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

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)

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_univariate¶

Bases: sage.rings.polynomial.laurent_polynomial.LaurentPolynomial

A univariate Laurent polynomial in the form of \(t^n \cdot f\) where \(f\) is a polynomial in \(t\).

INPUT:

parent – a Laurent polynomial ring
f – a polynomial (or something can be coerced to one)
n – (default: 0) an integer

AUTHORS:

Tom Boothby (2011) copied this class almost verbatim from laurent_series_ring_element.pyx, so most of the credit goes to William Stein, David Joyner, and Robert Bradshaw
Travis Scrimshaw (09-2013): Cleaned-up and added a few extra methods

coefficients()¶

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]

constant_coefficient()¶

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

degree()¶

Return the degree of self.

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

derivative(*args)¶

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.<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

dict()¶

Return a dictionary representing self.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: Q.<t> = LaurentPolynomialRing(R)
sage: f = (x^3 + y/t^3)^3 + t^2; f
y^3*t^-9 + 3*x^3*y^2*t^-6 + 3*x^6*y*t^-3 + x^9 + t^2
sage: f.dict()
{-9: y^3, -6: 3*x^3*y^2, -3: 3*x^6*y, 0: x^9, 2: 1}

exponents()¶

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]

factor()¶

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)

gcd(right)¶

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

integral()¶

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 \(1/t\) is \(\log(t)\), 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

inverse_of_unit()¶

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

is_constant()¶

Return whether this Laurent polynomial 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
sage: R(0).is_constant()
True
sage: R(42).is_constant()
True
sage: x.is_constant()
False
sage: (1/x).is_constant()
False

is_monomial()¶

Return True if self is a monomial; that is, if self is \(x^n\) for some integer \(n\).

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

is_square(root=False)¶

Return whether this Laurent polynomial is a square.

If root is set to True then return a pair made of the boolean answer together with None or a square root.

EXAMPLES:

sage: R.<t> = LaurentPolynomialRing(QQ)

sage: R.one().is_square()
True
sage: R(2).is_square()
False

sage: t.is_square()
False
sage: (t**-2).is_square()
True

Usage of the root option:

sage: p = (1 + t^-1 - 2*t^3)
sage: p.is_square(root=True)
(False, None)
sage: (p**2).is_square(root=True)
(True, -t^-1 - 1 + 2*t^3)

The answer is dependent of the base ring:

sage: S.<u> = LaurentPolynomialRing(QQbar)
sage: (2 + 4*t + 2*t^2).is_square()
False
sage: (2 + 4*u + 2*u^2).is_square()
True

is_unit()¶

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.

is_zero()¶

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

number_of_terms()¶

Return the number of non-zero coefficients of self.

Also called weight, hamming weight or sparsity.

EXAMPLES:

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - 1
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+1)^100
sage: f.number_of_terms()
101

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

polynomial_construction()¶

Return the polynomial and the shift in power used to construct the Laurent polynomial \(t^n u\).

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)

quo_rem(right_r)¶

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)

residue()¶

Return the residue of self.

The residue is the coefficient of \(t^-1\).

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

shift(k)¶

Return this Laurent polynomial multiplied by the power \(t^n\). 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

truncate(n)¶

Return a polynomial with degree at most \(n-1\) whose \(j\)-th coefficients agree with self for all \(j < n\).

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

valuation(p=None)¶

Return the valuation of self.

The valuation of a Laurent polynomial \(t^n u\) is \(n\) plus the valuation of \(u\).

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

variable_name()¶

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'

variables()¶

Return the tuple of variables occurring 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()
()

Elements of Laurent polynomial rings¶

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