Fewnomial Systems with Many Roots, and an Adelic Tau Conjecture

TL;DR

This paper investigates bounds on the number of roots of fewnomial systems over local fields, introduces explicit systems nearing these bounds, and explores their implications for computational complexity and number theory.

Contribution

It proposes conjecturally tight bounds on roots of fewnomial systems and constructs systems approaching these bounds, linking algebraic geometry with computational complexity.

Findings

Explicit systems with roots approaching known upper bounds

Connections established between root bounds and computational complexity

Construction of tropical varieties with maximal intersections

Abstract

Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the Shub-Smale tau-Conjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.