On the expected number of real roots of polynomials and exponential sums

TL;DR

This paper generalizes known results about the expected number of real roots in random polynomial systems to include certain sparse systems without requiring orthogonal invariance, expanding the theoretical understanding of root distributions.

Contribution

It extends previous findings to sparse polynomial systems, removing the assumption of orthogonal invariance and broadening the scope of expected root count analysis.

Findings

Expected number of real roots for sparse systems derived

Generalization of invariance assumptions achieved

Theoretical framework expanded for polynomial root analysis

Abstract

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the B\'ezout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.