An approach to singularity from the extended Hensel construction

Abstract

Let F(x,u), with (u) = (u1,...,uℓ), be an irreducible multivariate polynomial over C, having singularity at the origin, and let FNew(x,u) be the so-called Newton polynomial for F(x,u). The extended Hensel construction (EHC in short) of F(x,u) allows us to compute the Puiseux-series roots if ℓ = 1 and, for ℓ ≥ 2, the roots which are fractional-power series w.r.t. the (weighted) total-degree of u1,...,uℓ. This paper investigates the behavior of algebraic function χ(u) around the origin, where χ(u) is a root of F(x,u), w.r.t. the variable x, and clarifies a close relationship between the singularity of χ(u) and the corresponding Hensel factor. Let the irreducible factorization of FNew in C[x,u] be FNew(x,u)^m1...Hr(x,u)^m. It is shown that: 1) the procedure of EHC distributes the factors of the leading coefficient of F(x,u) to mutually prime factors of FNew in a unique way, and the "scaled-root" x(u) becomes infinity only at the zero-points of the leading coefficient of the corresponding factor of FNew(x,u); 2) the behavior of χ(u) around the origin changes singularly at zero-points of resultant(Hi, Hj) (∀i ≠ j), resultant(Hi,∂Hi/∂x) (i = 1,...,r), and fcirc;ℓ (ℓ = 1,...,ρ), where fℓ is the sum of terms plotted at the ℓ-th vertex of bottom sides of the "Newton polygon". We explain these points not only theoretically but also by examples.