On the Expected Number of Zeros of Nonlinear Equations

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 one 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.