A resultant for Hensel's Lemma

TL;DR

This paper generalizes Hensel's Lemma to factorisations into multiple factors over complete discrete valuation rings by defining a new resultant for several polynomials, enabling direct lifting of factorizations.

Contribution

It introduces a generalized resultant for multiple polynomials and extends Hensel's Lemma to handle multiple factors simultaneously, simplifying the factorization lifting process.

Findings

Provides a unique lifting of factorisations into several factors from mod pi^s to R[X]

Defines a new resultant as a determinant of a generalized Sylvester matrix

Avoids iterative two-factor factorizations by working directly with multiple factors

Abstract

Let R be a complete discrete valuation ring with maximal ideal generated by pi. Let f(X) in R[X] be a monic polynomial with nonzero discriminant Delta(f). Let s >= v_pi(Delta(f)) + 1. Suppose given a factorisation of f(X) in (R/pi^s R)[X] into several factors, not necessarily coprime in (R/pi R)[X]. We lift it uniquely to a factorisation of f(X) in R[X]. This generalises the Hensel-Rychlik Lemma, which covers the case of two factors. We work directly with lifts of factorisations into several factors and avoid having to iterate factorisations into two factors. For this purpose we define a resultant for several polynomials in one variable as determinant of a generalised Sylvester matrix.