Real zeros of mixed random fewnomial systems

Abstract

Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $f_i$ has a prescribed set of exponent vectors described by a set $A_i \subseteq \mathbb{Z}^n$ of cardinality $t_i$, whose convex hull is denoted $P_i$. Assuming that the coefficients of the $f_i$ are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most $(2\pi)^{-\frac{n}{2}} V_0 (t_1-1)\ldots (t_n-1)$. Here $V_0$ denotes the number of vertices of the Minkowski sum $P_1+\ldots + P_n$. However, this bound does not improve over the bound in B\"urgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019) for the unmixed case, where all supports $A_i$ are equal. All arguments equally work for real exponent vectors.