Estimating the p-adic valuation of the resultant

2021/11/11 by Kristof Szabo, Szabo, Kristof
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2111.06354

openalex publication_date 2021/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f and g be two monic polynomials with integer coefficients and nonzero resultant r. Assume that v_p(f(n))≥ s₁ and v_p(g(n))≥ s₂ hold for all integers n for some s₁, s₂ 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)≥ S-maxs₁,s₂+ps₁s₂\fracp-1p-p^-k, where k=\lfloor log_p((p-1)max\s₁,s₂\+1)\rfloor -1.

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