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The real tau-conjecture is true on average
Citation key BB-The-Real-Tau-Conjecture-Is-True-On-Average
Author Iréné Briquel and Peter Bürgisser
Year 2020
DOI 10.1002/rsa.20926
Journal Random Structures & Algortihms
Note This is an online version before inclusion in an issue of the journal.
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 $m,k,t$. 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 $O(mk^2t)$.
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