On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system [Formula: see text] of [Formula: see text] random real polynomials in [Formula: see text] variables, where each [Formula: see text] has a prescribed set of exponent vectors in a set [Formula: see text] of size [Formula: see text]. Assuming that the coefficients of the [Formula: see text] 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 [Formula: see text]. This result is a probabilistic version of Kushnirenko’s conjecture; it provides a bound that only depends on the number of terms and is independent of their degree.