Elementary analysis of isolated zeroes of a polynomial system

TL;DR

This paper extends Wooley's elementary proof of a Bezout-like theorem to bound the number of isolated roots of polynomial systems over arbitrary fields and modulo powers of a variable, generalizing previous results.

Contribution

It adapts Wooley's proof to polynomials over polynomial rings, providing bounds on isolated roots modulo t^s for any positive integer s.

Findings

Bound on isolated roots modulo t^s for polynomial systems

Extension of Wooley's theorem to arbitrary fields

Application to systems over arbitrary fields

Abstract

Wooley ({\em J. Number Theory}, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the proof to a slightly different setting. Specifically, we consider polynomials with coefficients from a polynomial ring $\mathbb{F}[t]$ for an arbitrary field $\mathbb{F}$ and give an upper bound on the number of isolated roots modulo $t^s$ for an arbitrary positive integer $s$. In particular, using $s=1$, we can bound the number of isolated roots of a system of polynomials over an arbitrary field $\mathbb{F}$.