An approach to singularity from the extended Hensel construction

Abstract

Let F ( x,u ), with ( u ) = ( u 1 ,..., u ℓ ), be an irreducible multivariate polynomial over C, having singularity at the origin, and let F New ( 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 u 1 ,..., u ℓ . 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 F New in C[ x,u ] be F New ( x,u ) m 1 ... H r ( 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 F New 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 F New ( x,u ); 2) the behavior of χ( u ) around the origin changes singularly at zero-points of resultant( H i , H j ) (∀ i ≠ j ), resultant( H i ,∂ H i /∂ x ) ( i = 1,..., r ), and fcirc; ℓ (ℓ = 1,...,ρ), where f ℓ 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.