Hensel construction of F(x, u1, ..., xl) l ≥ 2 at a singular point and its applications

• Authors: Tateaki Sasaki;Daiju Inaba
• Venue: ACM SIGSAM Bulletin
• Year: 2000

In 1993, Sasaki and Kako proposed a Hensel-like construction ofF(x,u1,¡­,ul),l¡Ý 2, at a singular point where the conventionalgeneralized Hensel construction breaks down. In this paper, wefirst extend Sasaki-Kako's method so as to apply to polynomialswith vanishing leading coefficients. Then, we investigate a specialcase that the initial factors are polynomials. We prove that themultivariate polynomial can be decomposed at any singular pointinto factors which are polynomials in a main variable withcoefficients being (infinite) series of rational functions suchthat¡Æk=0¡Þ[Nk(u1¡­,ul)/Dk(u1,¡­,ul)].Here, Nk and Dk are homogeneouspolynomials in u1 -s1,¡­,ul -sl and tdeg(Nk) -tdeg(Dk) = k, where(s1,¡­,sl) is theexpansion point and tdeg denotes the total-degree. Theextended Hensel construction can be used to factorization ofmultivariate polynomials having a singular point at the origin.After performing the extended Hensel construction at the origin, wesearch for the smallest subsets of Hensel factors such that theproduct of the members of each subset contains no rationalfunction. Then, we obtain the factorization inK{u1,¡­,ul}[x], with K a number field. Next, we search for thesmallest subsets such that the product of the members of eachsubset contains no infinite series. Then, we obtain thefactorization in K[x,u1,¡­,ul], withoutemploying the so-called nonzero substitution.