The real zeros of a random algebraic polynomial with dependent coefficients

TL;DR

This paper extends the analysis of the expected number of real zeros of random algebraic polynomials with dependent coefficients, showing that under certain spectral density conditions, the expectation remains similar to the independent case.

Contribution

It generalizes previous results by Sambandham to a broader class of covariance functions, demonstrating the robustness of the expected zero count.

Findings

Expected number of zeros matches the independent case under certain spectral conditions

Dependence structure can reduce the expected zeros by half in specific cases

Results apply to a wider class of covariance functions beyond exponential decay

Abstract

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of (2/pi)log(n), as n goes to infinity. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case looked at coefficients with an exponentially decaying covariance function, while the second assumed a constant covariance. He showed that the expectation of the number of real zeros for an exponentially decaying covariance matches the independent case, while having a constant covariance reduces the expected number of zeros in half. In this paper we will apply techniques similar to Sambandham's and extend his results to a wider class of covariance functions. Under certain restrictions on the spectral density, we will show that the order of the expected number of real zeros remains the same as in the independent case.