Root repulsion and faster solving for very sparse polynomials over p-adic fields

Abstract

TextFor any fixed field K∈{Q2,Q3,Q5,…}, we prove that all univariate polynomials f with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients in {±1,…,±H}, can be solved over K within deterministic time log4+o(1)⁡(dH)log3⁡d (resp. log2+o(1)⁡(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of f in K, and for each such root generates an approximation in Q with logarithmic height O(log2⁡(dH)log⁡d) that converges at a rate of O((1/p)2i) after i steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of p−O(plogp2⁡(dH)log⁡d) for distinct roots in Cp, and even stronger root repulsion when there are nonzero degenerate roots in Cp: p-adic distance p−O(logp⁡(dH)). On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in Zp indistinguishable in their first Ω(dlogp⁡H) most significant base-p digits. So speed-ups for t-nomials with t≥4 will require evasion or amortization of such worst-case instances. VideoFor a video summary of this paper, please visit https://youtu.be/npfdxLk04MY.