On the expected number of zeros of nonlinear equations

TL;DR

This paper analyzes the expected number of complex roots of analytic nonlinear equations within Gaussian-distributed function spaces, establishing a general theorem linking root counts of mixed and unmixed systems.

Contribution

It introduces a general theorem connecting the expected number of roots in mixed and unmixed systems of analytic equations within Gaussian spaces.

Findings

Root count is additive with respect to a product operation of functional spaces.

A general theorem relates expected roots of mixed and unmixed systems.

Examples include equations beyond polynomials and exponential sums.

Abstract

This paper investigates the expected number of complex roots of nonlinear equations. Those equations are assumed to be analytic, and to belong to certain inner product spaces. Those spaces are then endowed with the Gaussian probability distribution. The root count on a given domain is proved to be `additive' with respect to a product operation of functional spaces. This allows to deduce a general theorem relating the expected number of roots for unmixed and mixed systems. Examples of root counts for equations that are not polynomials nor exponential sums are given at the end.