Real $$\tau $$-Conjecture for Sum-of-Squares: A Unified Approach to Lower Bound and Derandomization

Abstract

Koiran’s real \(\tau \)-conjecture asserts that if a non-zero real univariate polynomial f can be written as \(\sum _{i=1}^{k}\prod _{j=1}^m\,f_{ij}\), where each \(f_{ij}\) contains at most t monomials, then the number of distinct real roots of f is polynomially bounded in kmt. Assuming the conjecture with parameter \(m=\omega (1)\), one can show that \({\mathsf {VP}}\ne {\mathsf {VNP}}\) (i.e. symbolic permanent requires superpolynomial-size circuit). In this paper, we propose a \(\tau \)-conjecture for sum-of-squares (SOS) model (equivalently, \(m=2\)).For a univariate polynomial f, we study the sum-of-squares representation (SOS), i.e. \(f = \sum _{i\in [s]} c_i f_i^2\) , where \(c_i\) are field elements and the \(f_i\)’s are univariate polynomials. The size of the representation is the number of monomials that appear across the \(f_i\)’s. Its minimum is the support-sum S(f) of f. We conjecture that any real univariate f can have at most O(S(f))-many real roots. A random polynomial satisfies this property. We connect this conjecture with two central open questions in algebraic complexity– matrix rigidity and \({\mathsf {VP}}\) vs. \({\mathsf {VNP}}\).The concept of matrix rigidity was introduced by Valiant (MFCS 1977) and independently by Grigoriev (1976) in the context of computing linear transformations. A matrix is rigid if it is far (in terms of Hamming distance) from any low rank matrix. We know that rigid matrices exist, yet their explicit construction is still a major open question. Here, we show that SOS-\(\tau \)-conjecture implies construction of such matrices. Moreover, the conjecture also implies the famous Valiant’s hypothesis (Valiant, STOC 1979) that \({\mathsf {VNP}}\) is exponentially harder than \({\mathsf {VP}}\). Thus, this new conjecture implies both the fundamental problems by Valiant.Furthermore, strengthening the conjecture to sum-of-cubes (SOC) implies that blackbox-PIT (Polynomial Identity Testing) is in \({\mathsf {P}}\). This is the first time a \(\tau \)-conjecture has been shown to give a polynomial-time PIT. We also establish some special cases of this conjecture, and prove tight lower bounds for restricted depth-2 models.Keywords\(\tau \)-conjectureMatrix rigidityReal root \({\mathsf {VP}}\) \({\mathsf {VNP}}\) PIT