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 .
Highlights
We study the number of real zeros of real univariate polynomials
A polynomial f is called t-sparse if it has at most t monomials
Descartes rule states that a t-sparse polynomial f has at most t−1 positive real zeros, no matter what is the degree of f
Summary
We study the number of real zeros of real univariate polynomials. A polynomial f is called t-sparse if it has at most t monomials. If we assume the coefficients uijs are independent random variables whose distributions have densities satisfying some mild assumptions, the expected number of real zeros of F in [0, 1] is bounded by a polynomial in k1 + · · · + km and t, provided 0 ∈ Sij for all i, j. The latter condition means that all the fij almost surely have a nonzero constant coefficient. It would be interesting to strengthen our result by concentration statements, showing that it is very unlikely that a random F of the above structure can have many real zeros
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