Ultrametric Root Counting

TL;DR

This paper introduces a reformulation of Hensel's lemma and a regularity condition that simplifies counting roots of polynomials over non-archimedean fields by relating it to roots of binomials from the Newton polygon.

Contribution

It provides a new reformulation of Hensel's lemma and introduces a regularity condition for straightforward root counting in non-archimedean fields.

Findings

Reformulation of Hensel's lemma relating roots of polynomials and their reductions.

Definition of a regularity condition for root counting.

Root count equals sum of roots of binomials from the Newton polygon.

Abstract

Let $K$ be a complete non-archimedean field with a discrete valuation, $f\in K[X]$ a polynomial with non-vanishing discriminant, $A$ the valuation ring of $K$, and $\M$ the maximal ideal of $A$. The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of $f$ with the number of roots of its reduction modulo a power of $\M$. We then define a condition --- {\em regularity} --- that yields a simple method to compute the exact number of roots of $f$ in $K$. In particular, we show that regularity implies that the number of roots of $f$ equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.