Counting Tropically Degenerate Valuations and p-adic Approaches to the Hardness of the Permanent

TL;DR

This paper explores new conjectures related to the hardness of the permanent, linking p-adic valuations of polynomial roots with complexity theory, and providing bounds on valuations of sparse polynomial systems.

Contribution

It introduces alternative conjectures to the Tau Conjecture that are potentially easier to prove and connects p-adic geometry with computational complexity.

Findings

New upper bounds on p-adic valuations of roots of sparse polynomials

Proposed alternative conjectures implying permanent hardness

Established links between p-adic geometry and complexity theory

Abstract

The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\neq NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some potentially easier to prove, whose truth still implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of $p$-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative p-adic geometry and complexity theory.