Estimating the $p$-adic valuation of the resultant

TL;DR

This paper derives new lower bounds for the p-adic valuation of the resultant of two monic integer polynomials, based on valuations of polynomial values and their gcd, advancing understanding of p-adic properties of resultants.

Contribution

It introduces multiple lower bounds for the p-adic valuation of the resultant, connecting valuations of polynomial evaluations and gcd in a novel way.

Findings

Established a lower bound for v_p(r) involving S, s_1, s_2, and p.

Derived explicit formula for the lower bound using p-adic logarithms.

Extended previous results by incorporating gcd valuations in the bounds.

Abstract

Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Assume that $v_p(f(n))\ge s_1$ and $v_p(g(n))\ge s_2$ hold for all integers $n$ for some $s_1, s_2$ fixed non-negative integers. Let $S$ denote the maximum of $v_p(gcd(f(n),g(n)))$ over all integers $n$. In this paper, we establish multiple lower bound for $v_p(r)$. More specifically, we show that $v_p(r)\ge S-\max{s_1,s_2}+ps_1s_2\frac{p-1}{p-p^{-k}}$, where $k=\lfloor \log_p((p-1)\max\{s_1,s_2\}+1)\rfloor -1$.