Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

Abstract

For any fixed field K ∈ {Q 2 , Q 3 , Q 5 , . . .}, we prove that all polynomials f ∈ Z[x] with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients having absolute value at most H, can be solved over K within deterministic time log 4+o(1) (dH) log 3 (d) (resp. log 2+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(log 2 (dH) log(d)) that converges at a rate of O (1/p) 2 i after i steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of p −O(p log 2 p (dH) log d) for distinct roots in C p , and even stronger repulsion when there are nonzero degenerate roots in C p : p-adic distance p −O(log p (dH)) . On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in Z p indistinguishable in their first Ω(d log p H) most significant base-p digits. Contentsp 2.4. Bit Complexity Basics and Counting Roots of Binomials 2.5. Trees and Roots in Z/(p k ) and Z p 2.6. Trees and Extracting Digits of Radicals 3.