doi:10.1155/2008/189675 Research Article On Different Classes of Algebraic Polynomials with Random Coefficients

The expected number of real zeros of the polynomial of the form a0 � a1xa2x 2 � ··· � anx n , where a0 ,a 1 ,a 2 ,...,a n is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in � −∞, ∞� is asymptotic to � 2/π� log n. In this paper, we show that this asymptotic value increases significantly to √ n � 1 when we consider a polynomial in the form a0 � n �1/2 x/ √ 1 � a1 � n � 1/2 x 2 / √ 2 � a2 � n �1/2 x 3 / √ 3 � ··· � ann �1/2 x n� 1 / √ n � 1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.


Introduction
The classical random algebraic polynomial has previously been defined as where, for Ω, A, Pr a fixed probability space, {a j ω } n j 0 is a sequence of independent random variables defined on Ω.For n large, the expected number of real zeros of T x , in the interval −∞, ∞ , defined by EN 0,T −∞, ∞ , is known to be asymptotic to 2/π log n.For this case the coefficients a j ≡ a j ω are assumed to be identical normal standard.This asymptotic value was first obtained by the pioneer work of Kac 1 and was recently significantly improved by Wilkins 2 , who reduced the error term involved in this asymptotic formula to O 1 .Since then, many other mathematical properties of T x have been studied and they are listed in 3 and more recently in 4 .
The other class of random polynomials is introduced in an interesting article of Edelman and Kostlan 5 in which the jth coefficients of T x in 1.1 have nonidentical variance n j .It is interesting to note that in this case the expected number of zeros significantly increased to √ n, showing that the curve representing this type of polynomial oscillates significantly more than the classical polynomial 1.1 with identical coefficients.As it is the characteristic of n j , j 0, 1, 2, . . ., n maximized at the middle term of j n/2 , it is natural to conjecture that for other classes of distributions with this property the polynomial will also oscillate significantly more.This conjecture is examined in 6, 7 .This interesting and unexpected property of the latter polynomial has its close relation to physics reported by Ramponi 8 , which together with its mathematical interest motivated us to study the polynomial As we will see, because of the presence of the binomial elements in 1.2 , we can progress further than the classical random polynomial defined in 1.1 .However, even in this case the calculation yields an asymptotic result rather than equality.With a small change to the definition of the polynomial we show that the result improves.To this end we define where a * is mutually independent of and has the same distribution as {a j } n j 0 .We prove the following.
Theorem 1.1.When the coefficients a j of P x are independent standard normal random variables, then the expected number of real roots is asymptotic to Corollary 1.2.With the same assumption as Theorem 1.1 for the coefficients a j and a * one has Also of interest is the expected number of times that a curve representing the polynomial cuts a level K.We assume K is any constant such that

1.6
For example, any absolute constant K / 0 satisfies these conditions.Defining EN K,P as the expected number of real roots of P x K, we can generalize the above theorem to the following one.

Theorem 1.3. When the coefficients a j have the same distribution as in Theorem 1.1, and K obeys the above conditions (i)-(iii), the asymptotic estimate for the expected number of K-level crossings is
The other characteristic which also gives a good indication of the oscillatory behavior of a random polynomial is the expected number of maxima or minima.We denote this expected number by EN M P for polynomial P x given in 1.2 and, since the event of tangency at the xaxis has probability zero, we note that this is asymptotically the same as the expected number of real zeros of P x dP x /dx.In the following theorem, we give the expected number of maxima of the polynomial.
Theorem 1.4.With the above assumptions on the coefficients a j , then the asymptotic estimate for the expected number of maxima of P x is Corollary 1.5.With the above assumptions for the coefficients a j and a * one has 1.9

Proof of Theorem 1.1
We use a well-known Kac-Rice formula, 1, 9 , in which it is proved that where P x represents the derivative with respect to x of P x .We denote Now, with our assumptions on the distribution of the coefficients, it is easy to see that We note that, for all sufficiently large n and x bounded away from zero, from 2.3 we have This together with 2.1 , 2.4 , and 2.5 yields where > 0, → 0 as n → ∞.The second integral can be expressed as In the first integral, the expression Δ/A 2 has a singularity at x 0:

2.9
Notice that the expression in 2.9 is bounded from above: where

2.11
When x 0, we have 12 and therefore which means that the integrand in the first integral of 2.7 is bounded for every n.When x > 0, it can easily be seen that 2.14 and therefore

2.15
Hence, the first integral that appears in 2.7 is bounded from above as follows: by the choice of .Altogether, the value of the first integral in 2.7 is of a smaller order of magnitude than the value of the second integral, and we have from 2.7 which completes the proof of Theorem 1.1.
In order to obtain the proof of Corollary 1.2, we note that the above calculations remain valid for B 2 and C.However, for A 2 we can obtain the exact value rather than the asymptotic value.To this end, we can easily see that Substituting this value instead of 2.3 together with 2.4 and 2.5 in the Kac-Rice formula 2.1 , we get a much more straight forward expression than that in the above proof:

2.19
This gives the proof of Corollary 1.2.

Level crossings
To find the expected number of K-level crossings, we use the following extension to the Kac-Rice formula as it was used in 10 .It is shown that in the case of normal standard distribution of the coefficients where, as usual, erf Hence to what follows we are only concerned with x ≥ 0. Using 2.3 -2.5 and 3.2 we obtain

Journal of Applied Mathematics and Stochastic Analysis
Using substitution x tan θ in 3.4 we can see that where the notation J 1 emphasizes integration in θ.In order to progress with the calculation of the integral appearing in 3.5 , we first assume θ > δ, where δ arccos 1 − 1/ n , where 1/{2 log nK }.This choice of is indeed possible by condition i .Now since cos θ < 1 − 1/ n , we can show that as n→∞.Now we are in a position to evaluate the dominated term which appears in the exponential term in 3.5 .From 3.6 , it is easy to see that for our choice of θ by condition ii .Therefore, for all sufficiently large n, the argument of the exponential function in 3.5 is reduced to zero, and hence the integrand is not a function of θ and we can easily see by the bounded convergence theorem and condition iii that Since the argument of the exponential function appearing in 3.5 is always negative, it is straight forward for our choice of δ and to see that by condition iii .As I 1 −∞, ∞ J 1 0, δ J 1 δ, π/2 , by 3.8 and 3.9 we see that

3.10
Now we obtain an upper limit for I 2 defined in 3.3 .To this end, we let v K/ √ 2A .Then we have

3.11
This together with 3.10 proves that EN Let us now prove the theorem for polynomial P x given in 1.2 , that is

3.12
The proof in this case repeats the proof for EN K,Q −∞, ∞ above, except that the equivalent of 3.4 will be an asymptotic rather than an exact equality, and the derivation of the equivalent of 3.9 is a little more involved, as shown below.Going back from the new variable θ to the original variable x gives where Δ/A 2 is given by 2.9 .Then by the same reasoning as in the proof of Theorem 1.1, 3.14 by condition iii .This completes the proof of Theorem 1.3.

Number of maxima
In finding the expected number of maxima of P x , we can find the expected number of zeros of its derivative P x .To this end we first obtain the following characteristics needed in order to apply them into the Kac-Rice formula 2.1 ,