Resultants and subresultants of p-adic polynomials

TL;DR

This paper investigates the stability of algorithms computing resultants and subresultants of p-adic polynomials, revealing their high instability on average and proposing stabilization methods without increasing complexity.

Contribution

It provides a thorough analysis of the stability issues in p-adic polynomial resultants and subresultants computations, and characterizes the distribution of valuations of subresultants.

Findings

Euclide-like algorithms are highly unstable on average

Methods to stabilize computations without increasing complexity

Distribution of valuations of subresultants for random polynomials

Abstract

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the distribution of the valuation of the principal subresultants of two random monic p-adic polynomials having the same degree.