The real tau-conjecture is true on average

TL;DR

This paper proves that Koiran's real tau-conjecture holds on average for polynomials with Gaussian coefficients, showing the expected number of real zeros is polynomially bounded, which impacts complexity theory.

Contribution

It establishes the average-case validity of the real tau-conjecture under Gaussian randomness, a significant step towards understanding polynomial zero bounds.

Findings

Expected number of real zeros is O(mk^2t) for Gaussian coefficients.

Confirms the conjecture in a probabilistic sense, not worst-case.

Implications for complexity theory and bounds on arithmetic circuit size.

Abstract

Koiran's real $\tau$-conjecture claims that the number of real zeros of a structured polynomial given as a sum of $m$ products of $k$ real sparse polynomials, each with at most $t$ monomials, is bounded by a polynomial in $m,k,t$. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of $f$ are independent standard Gaussian random variables, then the expected number of real zeros of $f$ is $O(mk^2t)$.