A Direct Ultrametric Approach to Additive Complexity and the Shub-Smale Tau Conjecture

TL;DR

This paper introduces a p-adic analogue approach to study the additive complexity of polynomials, providing new bounds on the number of roots and contributing to the understanding of the Shub-Smale Tau Conjecture, which links polynomial roots to computational complexity.

Contribution

The paper proves two weak versions of the Tau Conjecture and shows it follows from a more plausible hypothesis, using a novel p-adic approach to relate algebraic geometry and additive complexity.

Findings

Bound of O(e^{s log s}) roots in Q_2 for polynomials of additive complexity s

Improved upper bound on rational roots compared to previous real algebraic geometry methods

Progress towards an algorithmic arithmetic version of fewnomial theory

Abstract

The Shub-Smale Tau Conjecture is a hypothesis relating the number of integral roots of a polynomial f in one variable and the Straight-Line Program (SLP) complexity of f. A consequence of the truth of this conjecture is that, for the Blum-Shub-Smale model over the complex numbers, P differs from NP. We prove two weak versions of the Tau Conjecture and in so doing show that the Tau Conjecture follows from an even more plausible hypothesis. Our results follow from a new p-adic analogue of earlier work relating real algebraic geometry to additive complexity. For instance, we can show that a nonzero univariate polynomial of additive complexity s can have no more than 15+s^3(s+1)(7.5)^s s! =O(e^{s\log s}) roots in the 2-adic rational numbers Q_2, thus dramatically improving an earlier result of the author. This immediately implies the same bound on the number of ordinary rational roots, whereas the best previous upper bound via earlier techniques from real algebraic geometry was a quantity in Omega((22.6)^{s^2}). This paper presents another step in the author's program of establishing an algorithmic arithmetic version of fewnomial theory.