A Complexity Chasm for Solving Univariate Sparse Polynomial Equations Over p-adic Fields

Abstract

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field K∈ Q2,,Q3,Q5, …, we prove that any polynomial f Z [x] with exactly 3 monomial terms, degree d, and all coefficients having absolute value at most H, can be solved over K within deterministic time logO(1) (dH) in the classical Turing model. (The best previous algorithms were of complexity exponential in log d, even for just counting roots in Qp.) In particular, our algorithm generates approximations in Q with bit-length log O(1) (dH) to all the roots of f in K, and these approximations converge quadratically under Newton iteration. On the other hand, we give a unified family of tetranomials requiring Ω(d log H) digits to distinguish the base-p expansions of their roots in K.