On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system ${\mathfrak{f}}_1(x)=0,\dots ,{\mathfrak{f}}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each ${\mathfrak{f}}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq {\mathbb{Z}}^n$ of size $t_k$. Assuming that the coefficients of the ${\mathfrak{f}}_k$ are independent Gaussian of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by $4^{-n}{\prod }_{k=1}^n{t}_k(t_k-1)$. This result is a probabilisitc version of Kushnirenko’s conjecture; it provides a bound that only depends on the number of terms and is independent of their degree.