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The real tau‐conjecture is true on average

Briquel, Irénée; Bürgisser, Peter

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(mk2t).
Published in: Random Structures & Algorithms, 10.1002/rsa.20926, Wiley