On Hensel's roots and a factorization formula in Z[[x]]

TL;DR

This paper develops explicit formulas for Hensel lifts of polynomial roots modulo an odd prime and provides a factorization method over formal power series rings for specific reducible polynomials.

Contribution

It introduces new formulas for Hensel lifts and explicit factorizations in formal power series rings using Bell polynomials and Lagrange inversion.

Findings

Formulas for Hensel lifts modulo p

Explicit factorization over Z[[x]] for certain polynomials

Use of Bell polynomials and Lagrange inversion in derivations

Abstract

Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.