Real zeros of mixed random fewnomial systems

Abstract

Consider a system f 1 (x) = 0, . . . , fn(x) = 0 of n random real polynomials in n variables, where each f i has a prescribed set of exponent vectors in a set A i ⊆ 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 (2π) − n 2 V 0 (t 1 − 1) . . . (tn − 1). Here V 0 denotes the number of vertices of the Minkowski sum P 1 + . . . + Pn. We also derive a better bound in the unmixed case where all supports A i are equal, improving upon Bürgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019). All arguments equally work for real exponent vectors.