Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

Abstract

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n−1 x ⁿ⁻¹ with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 − |i − j|/n, for i = 0,…, n − 1 and j = 0,…, n − 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.

Keywords:

• Dependent random variables
• Kac–Rice formula
• Number of real zeros
• Random algebraic polynomials
• Real roots

Mathematics Subject Classification:

• Primary 60G99
• Secondary 60H99

This work was completed while the first author was visiting the Department of Mathematics at the University of Ulster. The hospitality of the Department of Mathematics at the University of Ulster was appreciated. The financial support for this visit was provided by the Research Institute of the University of Ulster.