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) = N−1 j=0 {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.


Introduction
Let {α j } n−1 j=1 and {β j } n−1 j=1 be sequences of independently normally distributed random variables with means zero and variances σ 2 .For a sequence of complex numbers η j = α j + iβ j , j = 1,2,...,n − 1, with η n ≡ η 0 ≡ 1, we define a (complex) random algebraic polynomial as η j z j . (1.1) Although there have been many results concerning real and complex roots of P n (z), most of them assume identical distributions for α j 's and β j 's, and therefore η j 's.These results, for the real case, are initiated by fundamental works of Kac [3] and Rice [1,2], and recently they are re-examined in an interesting work by Wilkins [4].The study of the mathematical behavior of P n (z), later generalized to the complex case first described by Ibragimov and Zeitouni [5] and then by Farahmand and Jahangiri [6], is reviewed in [7].
For physical applications and developments, we refer interested readers to [8] and the references therein.Further for the case of identical coefficients, Farahmand and Grigorash [9] study a case of nonzero means.However, in the study of random matrix theory, it turns out that a special form of P n (z), known as self-reciprocal random algebraic polynomial, is of interest in which the polynomial required, for all n and z, satisfies the relation P n (z) = z n P n (1/z).This yields a polynomial where η n ≡ η 0 ≡ 1, and η n− j is the complex conjugate of η j , j = 1,2,...,n − 1.
The assumption of η n ≡ η 0 ≡ 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.Our above form of P n (z) satisfies this requirement when The properties of zeros of reciprocal polynomials with deterministic coefficients are also discussed by Lakatos and Losonczi [10].

Random trigonometric polynomials
With simple transformation, for z = r exp (iθ), we can rewrite P n (z) in (1.1) as where For the above regrouping of terms, we used the self-reciprocating property of η j ≡ η n− j .Indeed, since for n even not all the coefficients η j in (1.1) can have a matching conjugate, the main interest, as far as random matrix theory is concerned, is for the case of n odd.The latter case, therefore, would be our main interest.However, in order to be complete, at this stage we present P n (z) for both n odd and n even.To this end, for n odd we have and for n even Now from (2.3), that is, for n odd, the polynomial of interest has the form where N = (n − 1)/2 and The classical form of random trigonometric polynomials is defined as The study of these types of random polynomials is initiated by Dunnage [11].Although Dunnage studied the actual number of real zeros, his work showed that, for N large, the expected number of real zeros of Q N (θ) is asymptotic to 2N/ √ 3.This result is later generalized to the case of a constant level crossing in [12] and to the case of nonstandard normal in [13,14].The other results concerning Q N (θ) can be found in [7].Further, as reported by Bharucha-Reid and Sambandham [15], Das [16] considered the expected number of zeros of a random trigonometric polynomial similar to (2.6).These types of random trigonometric polynomials, as we will see, have the advantage of being stationary with respect to θ.However, since we are interested in the expected number of zeros of P n (z) given in (2.3), we need to generalize the result to the number of level crossings of T N (θ).
In what follows, we therefore study the number of level crossings of T N (θ) with cos(N + θ/2).As this level is a function of θ, it can be seen as a moving level crossing case, where there is no known formula for its expected number of crossings with T N (θ).We will develop this in the following section.Denote this number in the interval (a,b) by ᏺ(a,b) and its expected value by Eᏺ(a,b).We prove the following theorem.
Theorem 2.1.With the above assumption on the distributions of α j s and β j s and for all sufficiently large N, (2.8)

Expected number of crossings
In what follows, we generalize the known result from constant level crossing to this moving level for the special form of T N (θ) given in (2.6).We use a formula known as the Kac-Rice formula which is originally derived for the expected number of real zeros (axes crossings) of a random algebraic polynomial.It is known (see, e.g., [7, page 12]) that the expected number of real zeros of polynomial T N (θ) in (2.3)  where ϕ(x 1 ,x 2 ) is the joint probability density function of T N (θ) + cos(N + θ/2) and its derivative T N (θ) − (1/2)sin(N + θ/2).Let and let C N (θ) ≡ C be the covariance of T N (θ) and T N (θ).Since α j 's and β j 's are independent in themselves and from each other, we can easily show that C = 0. Using the assumption of normality of the coefficients of the polynomial, we therefore obtain the required joint probability density function as This will enable us to evaluate a part of the Kac-Rice formula given in (3.1) as Now we let t = y/(B √ 2), which enables us to proceed with the above integration as (3.6)K. Farahmand 5 However, from the above definition of J(λ), it is easy to see that where erf(x) = x 0 exp (−u 2 )du.In the derivation of (3.7), use has been made of the following:  (3.12) Note that the first equality for Eᏺ(0,2π) in the above formula as well as (3.11) is valid for all N, which is a much stronger result than the one we stated here.However, the gain in stating such an untidy result does not justify the advantage of the generalization.
in the interval (a,b) is given by Eᏺ(a,b) = (0, y) dy, (3.1) 4 Journal of Applied Mathematics and Stochastic Analysis