Estimating the $p$-adic valuation of the resultant

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 $v_p(r)\ge S-\max s_1,s_2+p{s}_1{s}_2\frac{p-1}{p-p^{-k}}$, where $k=\lfloor {\log }_p((p-1)\max \{s_1,s_2\}+1)\rfloor -1$.

References in corpus (1)

• Estimating the greatest common divisor of the value of two polynomials