A complexity chasm for solving univariate sparse polynomial equations over $p$-adic fields

TL;DR

This paper demonstrates a sharp complexity divide in solving univariate sparse polynomial equations over p-adic fields, showing efficient algorithms for trinomials and inherent difficulty for tetranomials.

Contribution

It establishes a polynomial-time algorithm for solving trinomial equations over p-adic fields and proves a complexity lower bound for tetranomials, revealing a complexity chasm.

Findings

Polynomial-time solution for trinomial equations over p-adic fields.

Exponential complexity for tetranomials in root distinction.

Quadratic convergence of root approximations via Newton iteration.

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 $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $O(\log^{O(1)}(dH))$ in the classical Turing model. (The best previous algorithms were of complexity exponential in $\log d$, even for just counting roots in $\mathbb{Q}_p$.) In particular, our algorithm generates approximations in $\mathbb{Q}$ with bit-length $O(\log^{O(1)}(dH))$ 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 $\Omega(d\log H)$ digits to distinguish the base-$p$ expansions of their roots in $K$.