On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a polynomial system $f_1(x)=0,\dots ,f_n(x)=0$ of $n$ real polynomials in $n$ variables, where each $f_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq {\mathbb{Z}}^n$ of size $t_k$. Kushnirenko's conjecture asks whether the number of positive zeros of such a polynomial system is of the form $\mathrm{poly}(t_1,\dots ,t_n)$. As of today, this is one of the most challenging problems in real algebraic geometry, being open even in the two-dimensional case!
In this talk, we will present a probabilistic version of Kushnirenko's conjecture. We will show, among other variations of the result, that if the coefficients are independent Gaussian of any variance, then the expected number of positive zeros of the random system is bounded from above by $4^{-n}\prod _{k=1}^n{t}_k(t_k-1)$.
This is joint work with Alperen A. Ergür and Maté T. Telek. Preprint available at arxiv.org/abs/2306.06784