Real Zeros of Random Mixed Fewnomial System: The 'Trick' Strikes Back

Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ 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 $4^{-n} \prod_{k=1}^n t_k(t_k-1)$. Our results hold with much more flexible assumptions than any available result in the literature, and yield stronger estimates in the specific cases considered earlier by (B\"urgisser, 2023) [arXiv:2301.00273] and (B\"urgisser, Erg\"ur, Tonelli-Cueto, 2019) [arXiv:1811.09425]. In this sense, our results mark a new milestone in flexibility for the theory of random real fewnomials.