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

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 padic 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 O(d 2 log 3 d)-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.

Ultrametric Smale's $α$-theory

“…Proposition 2 (Ultrametric separation theorem for γ). [5,Theorem 3.15] Fix an algebraic closure F of F with the corresponding extension of the ultranorm. Let…”

Ultrametric smale's α-theory

“…In the p-adic setting, Breiding [1] proved a version of the γ-theorem, but he didn't provide a full α-theory. In this short communication, we provide an ultrametric version of Smale's α-theory for square systems-initially presented as an appendix in [5]-, together with an easy proof.…”