On Zeros of Self-Reciprocal Random Algebraic Polynomials

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial T N ( θ ) = ∑ j = 0 N − 1 { α N − j cos ( j + 1 / 2 ) θ + β N − j sin ( j + 1 / 2 ) θ } , where α j and β j , j = 0 , 1 , 2 , … , N − 1 , are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos ( N + θ / 2 ) . We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.