The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Bruno Grenet, Pascal Koiran, Natacha Portier & Yann Strozecki
Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real tau-conjecture" which is inspired by this connection. The real tau-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower...