Abstract
This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, 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.
Highlights
Let {α j }nj=−11 and {β j}n−1 j=1 be sequences of independently normally distributed random variables with means zero and variances σ2
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)
Dunnage studied the actual number of real zeros, his work show√ed that, for N large, the expected number of real zeros of QN (θ) is asymptotic to 2N/ 3
Summary
}n−1 j=1 be sequences of independently normally distributed random variables with means zero and variances σ2. There have been many results concerning real and complex roots of Pn(z), most of them assume identical distributions for αj’s and βj’s, and η j’s. Pn(z), known as self-reciprocal random algebraic polynomial, is of interest in which the polynomial required, for all n and z, satisfies the relation Pn(z) = znPn(1/z). This yields a polynomial where ηn ≡ η0 ≡ 1, and ηn−j is the complex conjugate of η j, j = 1, 2, . The assumption of ηn ≡ η0 ≡ 1 is motivated by the requirement that in the random matrix theory we are interested in polynomials whose (complex) zeros are located in the unit circle. The properties of zeros of reciprocal polynomials with deterministic coefficients are discussed by Lakatos and Losonczi [10]
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