Elements of Laurent polynomial rings¶
- class sage.rings.polynomial.laurent_polynomial.LaurentPolynomial¶
Bases:
sage.structure.element.CommutativeAlgebraElementBase 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
dictmethod.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
fto the coefficients ofself.If
fis asage.categories.map.Map, then the resulting polynomial will be defined over the codomain off. Otherwise, the resulting polynomial will be over the same ring asself. Setnew_base_ringto override this behavior.INPUT:
f– a callable that will be applied to the coefficients ofself.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.LaurentPolynomialMultivariate Laurent polynomials.
- coefficient(mon)¶
Return the coefficient of
moninself, wheremonmust have the same parent asself.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
selfin 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
xinself.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
selfrepresented as adict.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 ofself.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
Trueifselfis 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 (defaultFalse) - if set toTruethen return a pair(True, sqrt)withsqrta 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
Trueifselfis 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
Trueif this is a univariate or constant Laurent polynomial, andFalseotherwise.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
selfas 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
moninself, wheremonmust have the same parent asself.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
rightand 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– adictwhose 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 rescalingnew_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
his 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
his 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
ValueErrorexception is raised. The new polynomial is over the same base ring as the givenLaurentPolynomialand in the variablexif no ringRis 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.LaurentPolynomialA univariate Laurent polynomial in the form of \(t^n \cdot f\) where \(f\) is a polynomial in \(t\).
INPUT:
parent– a Laurent polynomial ringf– 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 BradshawTravis 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
selfwith 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
selfwithrightwhere the common divisordmakes bothselfandrightinto 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
selfif 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
Trueifselfis a monomial; that is, ifselfis \(x^n\) 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
rootis set toTruethen return a pair made of the boolean answer together withNoneor 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
rootoption: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
Trueif 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
1ifselfis 0, else return0.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)whereuis the underlying polynomial andnis 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
selfbyrightand 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
selffor 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
selfas 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() ()