A Wronskian approach to the real τ-conjecture

Abstract

According to the real τ -conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent.In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be repeated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials.Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).

Fewnomial systems with many roots, and an Adelic Tau Conjecture

“…So we pose our conjecture mainly to advocate adding p-adic techniques to the real-analytic toolbox put forth in [Koi11, Sec. 6] and [KPT12].…”

Intersection multiplicity of a sparse curve and a low-degree curve

“…In this section, we present two lemmas about derivatives that are used in the proof of our main theorem. These results appeared in [KPT15a,KPT15b] in the context of analytic functions and they carry over to Puiseux series. We use the convention that N = {0, 1, .…”

Factoring bivariate lacunary polynomials without heights

“…Our proof of Theorem 1 is based on the so-called Wronskian of a family of polynomials. This is a classical tool for the study of differential equations but it has recently been used to bound the valuation of a sum of square roots of polynomials [16] and also to bound the number of real roots of some sparse-like polynomials [18].…”