Hensel construction of F(x, u 1 , ..., x l ) l ≥ 2 at a singular point and its applications
Tateaki Sasaki, Daiju Inaba · ACM SIGSAM Bulletin · 2000
In 1993, Sasaki and Kako proposed a Hensel-like construction of F ( x, u 1 ,¡, u l ), l ¡Ý 2, at a singular point where the conventional generalized Hensel construction breaks down. In this paper, we first extend Sasaki-Kako's method so as to apply to polynomials with vanishing leading coefficients. Then, we investigate a special case that the initial factors are polynomials. We prove that the multivariate polynomial can be decomposed at any singular point into factors which are polynomials in a main variable with coefficients being (infinite) series of rational functions such that ¡Æ k =0 ¡Þ [ N k ( u 1 ¡, u l )/ D k ( u 1 ,¡, u l )]. Here, N k and D k are homogeneous polynomials in u 1 - s 1 ,¡, u l - s l and tdeg( N k ) - tdeg( D k ) = k, where ( s 1 ,¡, s l ) is the expansion point and tdeg denotes the total-degree. The extended Hensel construction can be used to factorization of multivariate polynomials having a singular point at the origin. After performing the extended Hensel construction at the origin, we search for the smallest subsets of Hensel factors such that the product of the members of each subset contains no rational function. Then, we obtain the factorization in K { u 1 ,¡, u l }[ x ], with K a number field. Next, we search for the smallest subsets such that the product of the members of each subset contains no infinite series. Then, we obtain the factorization in K [ x, u 1 ,¡, u l ], without employing the so-called nonzero substitution.