Elements of Laurent polynomial rings

Elements of Laurent polynomial rings

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_generic

Bases: sage.structure.element.CommutativeAlgebraElement

A generic Laurent polynomial.

class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair

Bases: sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_generic

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 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
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()

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)]
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()

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
is_monomial()

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
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
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
univariate_polynomial(R=None)

Returns a univariate polynomial associated to this multivariate polynomial.

INPUT:

  • R - (default: None) PolynomialRing

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_generic

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
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
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 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
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 :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
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 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
is_monomial()

Return True if this element 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_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
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 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()
()

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