Real zeros of mixed random fewnomial systems
Peter B\"urgisser

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 of random real polynomials in variables, where each has a prescribed set of exponent vectors described by a set of cardinality , whose convex hull is denoted . Assuming that the coefficients of the are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most . Here denotes the number of vertices of the Minkowski sum . 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 are equal. All arguments equally work for real exponent vectors.
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Taxonomy
TopicsGeometry and complex manifolds · Stochastic processes and statistical mechanics · Mathematical Dynamics and Fractals
