The real tau‐conjecture is true on average. Issue 2 (15th May 2020) - Briquel, Irénée; Bürgisser, Peter
Abstract : Koiran's real τ ‐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 mkt . This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower 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 ( m k 2 t ) .