Real zeros of mixed random fewnomial systems

TL;DR

This paper establishes an upper bound on the expected number of real zeros in the positive orthant for systems of random polynomial equations with prescribed supports, extending previous bounds to mixed systems with real exponents.

Contribution

It provides a new bound on the expected zeros of mixed random polynomial systems with real exponents, generalizing previous results for unmixed systems.

Findings

Expected zeros bound: at most $(2\pi)^{-rac{n}{2}} V_0 (t_1-1) imes\ldots imes (t_n-1)$

Bound matches previous bounds in the unmixed case

Arguments apply to real exponent vectors as well

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.