NewtonPuiseux(K, var, cen)

newtonpuiseux.spad line 252 [edit on github]

• K: Join(AlgebraicallyClosedField, PolynomialFactorizationExplicit)

• var: Symbol

• cen: K

Implements the computation of a Puiseux series satisfying a bivariate polynomial.

coefficientRelation: (SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer), List Point Fraction Integer) -> SparseUnivariatePolynomial K

coefficientRelation(p, points) returns the relation that the coefficient must fulfil in order to be a coefficient of the puiseux series.

coerce: PolynomialRing(K, Fraction Integer) -> UnivariatePuiseuxSeries(K, var, cen)

elt: (SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer), UnivariatePuiseuxSeries(K, var, cen)) -> UnivariatePuiseuxSeries(K, var, cen)

p(s) replaces the variable in the univariate polynomial ring by the Puiseux series s.

leadingPuiseuxMonomials: (Polynomial K, Symbol, Symbol) -> List Record(k: Fraction Integer, c: K)

leadingPuiseuxMonomials(p, x, y) returns for each slope of the Newton polygon of p the leading monomial of the Puiseux series Y satisfying p(x, Y(x)) = 0 in terms of the defining polynomial for the leading coefficient, i.e. one monomial might correspond to several coefficients according to the degree of the defining polynomial.

leadingPuiseuxMonomials: (SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer), Fraction Integer) -> List Record(k: Fraction Integer, c: K)

leadingPuiseuxMonomials: SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer) -> List Record(k: Fraction Integer, c: K)

leadingPuiseuxMonomials(p) returns for each slope of the Newton polygon of p the leading monomial of the Puiseux series Y satisfying p(x, Y(x)) = 0 in terms of the defining polynomial for the leading coefficient, i.e. one monomial might correspond to several coefficients according to the degree of the defining polynomial.

leadingPuiseuxMonomials: SparseUnivariatePolynomial SparseUnivariatePolynomial K -> List Record(k: Fraction Integer, c: K)

leadingPuiseuxMonomials(p) returns for each slope of the Newton polygon of p the leading monomial of the Puiseux series Y satisfying p(x, Y(x)) = 0 in terms of the defining polynomial for the leading coefficient, i.e. one monomial might correspond to several coefficients according to the degree of the defining polynomial.

nextPuiseuxMonomials: PreNewtonPuiseux K -> PreNewtonPuiseux K

nextY: (SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer), Record(k: Fraction Integer, c: K)) -> SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer)

poly2KPXY: (Polynomial K, Symbol, Symbol) -> SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer)

polynomial2Points: (Polynomial K, Symbol, Symbol) -> List Point Fraction Integer

polynomial2Points: SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer) -> List Point Fraction Integer

polynomial2Points: SparseUnivariatePolynomial SparseUnivariatePolynomial K -> List Point Fraction Integer

puiseuxSolutions: SparseUnivariatePolynomial PolynomialRing(K, Fraction Integer) -> List UnivariatePuiseuxSeries(K, var, cen)

puiseuxSolutions(p) returns degree(p) Puiseux series solutions s such that p(s)=0.