A $p$-adic Descartes solver: the Strassman solver

TL;DR

This paper introduces a $p$-adic version of Descartes' rule of signs, called Strassman's theorem, and develops an efficient algorithm for isolating roots of $p$-adic polynomials, with complexity analysis using a new framework.

Contribution

It translates Descartes' rule of signs into the $p$-adic setting and provides an efficient root isolation algorithm with complexity analysis in $p$-adic algebraic geometry.

Findings

Algorithm runs in $ ilde{O}(d^2)$ time for random degree $d$ polynomials.

Introduces a condition-based complexity framework for $p$-adic algebraic geometry.

Successfully adapts real root isolation techniques to the $p$-adic context.

Abstract

Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the $p$-adic worlds. We show how the $p$-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free $p$-adic polynomial. Moreover, we show that this algorithm runs in $\mathcal{O}(d^2\log^3d)$-time for a random $p$-adic polynomial of degree $d$. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into $p$-adic numerical algebraic geometry.