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

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