Table of Contents
Journal of Applied Mathematics and Stochastic Analysis
VolumeΒ 2008, Article IDΒ 189675, 8 pages
Research Article

On Different Classes of Algebraic Polynomials with Random Coefficients

Department of Mathematics, University of Ulster, Jordanstown, County Antrim BT37 0QB, Northern Ireland, UK

Received 27 February 2008; Accepted 20 April 2008

Academic Editor: LevΒ Abolnikov

Copyright Β© 2008 K. Farahmand et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The expected number of real zeros of the polynomial of the form π‘Ž0+π‘Ž1π‘₯+π‘Ž2π‘₯2+β‹―+π‘Žπ‘›π‘₯𝑛, where π‘Ž0,π‘Ž1,π‘Ž2,…,π‘Žπ‘› is a sequence of standard Gaussian random variables, is known. For 𝑛 large it is shown that this expected number in (βˆ’βˆž,∞) is asymptotic to (2/πœ‹)log𝑛. In this paper, we show that this asymptotic value increases significantly to βˆšπ‘›+1 when we consider a polynomial in the form π‘Ž0𝑛01/2√π‘₯/1+π‘Ž1𝑛11/2π‘₯2/√2+π‘Ž2𝑛21/2π‘₯3/√3+β‹―+π‘Žπ‘›ξ‚€π‘›π‘›ξ‚1/2π‘₯𝑛+1/βˆšπ‘›+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.

1. Introduction

The classical random algebraic polynomial has previously been defined as𝑇(π‘₯)≑𝑇𝑛(π‘₯,πœ”)=𝑛𝑗=0π‘Žπ‘—(πœ”)π‘₯𝑗,(1.1)where, for (Ξ©,π’œ,Pr) a fixed probability space, {π‘Žπ‘—(πœ”)}𝑛𝑗=0 is a sequence of independent random variables defined on Ξ©. For 𝑛 large, the expected number of real zeros of 𝑇(π‘₯), in the interval (βˆ’βˆž,∞), defined by EN0,𝑇(βˆ’βˆž,∞), is known to be asymptotic to (2/πœ‹)log𝑛. For this case the coefficients π‘Žπ‘—β‰‘π‘Žπ‘—(πœ”) 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(π‘₯) 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(π‘₯) in (1.1) have nonidentical variance (𝑛𝑗). It is interesting to note that in this case the expected number of zeros significantly increased to βˆšπ‘›, 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 (𝑛𝑗), 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 polynomial𝑃(π‘₯)≑𝑃𝑛(π‘₯,πœ”)=𝑛𝑗=0π‘Žπ‘—ξƒ©π‘›π‘—ξƒͺ1/2π‘₯𝑗+1βˆšπ‘—+1.(1.2)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 define𝑄(π‘₯)≑𝑄𝑛(π‘₯,πœ”)=𝑛𝑗=0π‘Žπ‘—ξƒ©π‘›π‘—ξƒͺ1/2π‘₯𝑗+1√+π‘Žπ‘—+1βˆ—βˆšπ‘›+1,(1.3)where π‘Žβˆ— is mutually independent of and has the same distribution as {π‘Žπ‘—}𝑛𝑗=0. We prove the following.

Theorem 1.1. When the coefficients π‘Žπ‘— of 𝑃(π‘₯) are independent standard normal random variables, then the expected number of real roots is asymptotic to 𝐸𝑁0,π‘ƒβˆš(βˆ’βˆž,∞)βˆΌπ‘›+1.(1.4)

Corollary 1.2. With the same assumption as Theorem 1.1 for the coefficients π‘Žπ‘— and π‘Žβˆ— one has 𝐸𝑁0,π‘„βˆš(βˆ’βˆž,∞)=𝑛+1.(1.5)

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(i)𝐾2≀𝑒𝑛𝑛2,1(ii)𝑛2𝐾=π‘œ2,(iii)𝐾2𝑒=π‘œπ‘›π‘›2.(1.6)For example, any absolute constant 𝐾≠0 satisfies these conditions. Defining EN𝐾,𝑃 as the expected number of real roots of 𝑃(π‘₯)=𝐾, we can generalize the above theorem to the following one.

Theorem 1.3. When the coefficients π‘Žπ‘— 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 𝐸𝑁𝐾,𝑄(βˆ’βˆž,∞)βˆΌπΈπ‘πΎ,π‘ƒβˆš(βˆ’βˆž,∞)βˆΌπ‘›+1.(1.7)

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 EN𝑀𝑃 for polynomial 𝑃(π‘₯) given in (1.2) and, since the event of tangency at the π‘₯-axis has probability zero, we note that this is asymptotically the same as the expected number of real zeros of π‘ƒξ…ž(π‘₯)=𝑑𝑃(π‘₯)/𝑑π‘₯. In the following theorem, we give the expected number of maxima of the polynomial.

Theorem 1.4. With the above assumptions on the coefficients π‘Žπ‘—, then the asymptotic estimate for the expected number of maxima of 𝑃(π‘₯) is πΈπ‘π‘€π‘ƒβˆš(βˆ’βˆž,∞)βˆΌπ‘›.(1.8)

Corollary 1.5. With the above assumptions for the coefficients π‘Žπ‘— and π‘Žβˆ— one has πΈπ‘π‘€π‘„βˆš(βˆ’βˆž,∞)βˆΌπ‘›.(1.9)

2. Proof of Theorem 1.1

We use a well-known Kac-Rice formula, [1, 9], in which it is proved thatEN0,π‘ƒξ€œ(π‘Ž,𝑏)=π‘π‘ŽΞ”πœ‹π΄2𝑑π‘₯,(2.1)where π‘ƒξ…ž(π‘₯) represents the derivative with respect to π‘₯ of 𝑃(π‘₯). We denote𝐴2=var𝑃(π‘₯),𝐡2𝑃=varξ…žξ‚ξ‚€(π‘₯),𝐢=cov𝑃(π‘₯),π‘ƒξ…žξ‚(π‘₯),Ξ”2=𝐴2𝐡2βˆ’πΆ2.(2.2)Now, with our assumptions on the distribution of the coefficients, it is easy to see that𝐴2=𝑛𝑗=0𝑛𝑗ξƒͺπ‘₯2𝑗+2=𝑗+11+π‘₯2𝑛+1βˆ’1𝑛+1,𝐡𝑛+1(2.3)2=𝑛𝑗=0𝑛𝑗ξƒͺ(𝑗+1)π‘₯2𝑗=ξ‚€1+π‘₯2ξ‚π‘›βˆ’1ξ‚€1+π‘₯2+𝑛π‘₯2,(2.4)𝐢=𝑛𝑗=0𝑛𝑗ξƒͺπ‘₯2𝑗+1ξ‚€=π‘₯1+π‘₯2𝑛.(2.5) We note that, for all sufficiently large 𝑛 and π‘₯ bounded away from zero, from (2.3) we have𝐴2βˆΌξ‚€1+π‘₯2𝑛+1𝑛+1.(2.6)This together with (2.1), (2.4), and (2.5) yieldsEN02(βˆ’βˆž,∞)βˆΌπœ‹ξ€œπœ–0Δ𝐴22𝑑π‘₯+πœ‹ξ€œβˆžπœ–βˆšπ‘›+11+π‘₯2𝑑π‘₯,(2.7)where πœ–>0, πœ–β†’0 as π‘›β†’βˆž. The second integral can be expressed as2βˆšπ‘›+1πœ‹ξ‚†πœ‹2ξ‚‡β†’βˆšβˆ’arctanπœ–π‘›+1asπ‘›β†’βˆž.(2.8)In the first integral, the expression (Ξ”/𝐴2) has a singularity at π‘₯=0:Δ𝐴2=ξ„Άξ„΅ξ„΅ξ„΅ξ„΅ξ„΅βŽ·(𝑛+1)1+π‘₯22π‘›βˆ’ξ‚€1+π‘₯2ξ‚π‘›βˆ’1ξ‚€1+π‘₯2+𝑛π‘₯21+π‘₯2𝑛+1ξ‚‡βˆ’12.(2.9)Notice that the expression in (2.9) is bounded from above:Δ𝐴2<√(𝑛+1)(1βˆ’π·)1+π‘₯2,(2.10)where𝐷=1+𝑛π‘₯2ξ‚€1+π‘₯2ξ‚π‘›βˆ’1βˆ’ξ‚€1+π‘₯2𝑛1+π‘₯2ξ‚π‘›ξ‚‡βˆ’12=ξ‚€(π‘›βˆ’1)1+π‘₯2ξ‚π‘›βˆ’2ξ‚€+(π‘›βˆ’2)1+π‘₯2ξ‚π‘›βˆ’3ξ‚€+β‹―+31+π‘₯22ξ‚€+21+π‘₯2+11+π‘₯2ξ‚π‘›βˆ’1+ξ‚€1+π‘₯2ξ‚π‘›βˆ’2ξ‚€+β‹―+1+π‘₯2+12.(2.11) When π‘₯=0, we have𝑛𝐷=2βˆ’π‘›2𝑛2(2.12)and therefore Δ𝐴2<𝑛+1βˆšβˆΌξ‚™2𝑛𝑛+12,(2.13)which means that the integrand in the first integral of (2.7) is bounded for every 𝑛. When π‘₯>0, it can easily be seen thatβˆ‘1>𝐷>π‘›βˆ’2𝑗=0(1+𝑗)𝑛2ξ‚€1+π‘₯22π‘›βˆ’2>0,(2.14)and thereforeΔ𝐴2<βˆšπ‘›+11+π‘₯2.(2.15)Hence, the first integral that appears in (2.7) is bounded from above as follows:2πœ‹ξ€œπœ–0Δ𝐴22𝑑π‘₯<πœ‹ξ€œπœ–0βˆšπ‘›+11+π‘₯2βˆšπ‘‘π‘₯=2(arctanπœ–)𝑛+1πœ‹ξ€·βˆš=π‘œξ€Έπ‘›+1(2.16)by the choice of πœ–. 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)EN0√(βˆ’βˆž,∞)βˆΌπ‘›+1(2.17)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 𝐡2 and 𝐢. However, for 𝐴2 we can obtain the exact value rather than the asymptotic value. To this end, we can easily see that𝐴2𝑄==var𝑄(π‘₯)𝑛𝑗=0𝑛𝑗ξƒͺπ‘₯2𝑗+2+1𝑗+1=𝑛+11+π‘₯2𝑛+1𝑛+1.(2.18)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:EN0,𝑄1(βˆ’βˆž,∞)=πœ‹ξ€œβˆž0βˆšπ‘›+11+π‘₯2βˆšπ‘‘π‘₯=𝑛+1.(2.19)This gives the proof of Corollary 1.2.

3. 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 coefficientsEN𝐾(π‘Ž,𝑏)=𝐼1(π‘Ž,𝑏)+𝐼2(π‘Ž,𝑏)(3.1)with 𝐼1ξ€œ(π‘Ž,𝑏)=π‘π‘ŽΞ”πœ‹π΄2ξ‚€βˆ’π΅exp2𝐾22Ξ”2𝐼𝑑π‘₯,(3.2)2ξ€œ(π‘Ž,𝑏)=π‘π‘Žβˆš2πΎπΆπœ‹π΄3ξ‚€βˆ’πΎexp22𝐴2ξ‚ξ‚€βˆ’erfπΎπΆβˆšξ‚2𝐴Δ𝑑π‘₯,(3.3) where, as usual, ∫erf(π‘₯)=π‘₯0√exp(βˆ’π‘‘)π‘‘π‘‘β‰€πœ‹/2. Since changing π‘₯ to βˆ’π‘₯ leaves the distribution of the coefficients unchanged, EN𝐾(βˆ’βˆž,0)=EN𝐾(0,∞). Hence to what follows we are only concerned with π‘₯β‰₯0. Using (2.3)–(2.5) and (3.2) we obtain𝐼12√(βˆ’βˆž,∞)=𝑛+1πœ‹ξ€œβˆž011+π‘₯2ξ‚€βˆ’πΎexp2ξ‚€(𝑛+1)1+π‘₯2+𝑛π‘₯22ξ‚€1+π‘₯2𝑛+1𝑑π‘₯.(3.4)Using substitution π‘₯=tanπœƒ in (3.4) we can see that𝐼1(βˆ’βˆž,∞)=𝐽1ξ‚€πœ‹0,2=2βˆšπ‘›+1πœ‹ξ€œ0πœ‹/2ξ‚€expβˆ’πΎ2(𝑛+1)2ξ‚€1+𝑛sin2πœƒξ‚cos2π‘›πœƒξ‚π‘‘πœƒ,(3.5)where the notation 𝐽1 emphasizes integration in πœƒ. In order to progress with the calculation of the integral appearing in (3.5), we first assume πœƒ>𝛿, where 𝛿=arccos(1βˆ’1/(π‘›πœ–)), where πœ–=1/{2log(𝑛𝐾)}. This choice of πœ– is indeed possible by condition (i). Now since cosπœƒ<(1βˆ’1/(π‘›πœ–)), we can show thatcos2𝑛1πœƒ<1βˆ’ξ‚π‘›πœ–2𝑛=1ξ‚€ξ‚€1βˆ’ξ‚π‘›πœ–βˆ’π‘›πœ–ξ‚βˆ’2/πœ–ξ‚€βˆ’2∼expπœ–ξ‚β†’0(3.6)as π‘›β†’βˆž. 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 πœƒπΎ2𝑛2cos2π‘›βˆ’2πœƒ<𝐾2𝑛2ξ‚€βˆ’2expπœ–ξ‚=𝐾2𝑛2expβˆ’4log(𝑛𝐾)=(𝐾𝑛)βˆ’2β†’0,(3.7)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 πœƒ and we can easily see by the bounded convergence theorem and condition (iii) that𝐽1ξ‚€πœ‹π›Ώ,2ξ‚βˆΌβˆšπ‘›+1.(3.8)Since the argument of the exponential function appearing in (3.5) is always negative, it is straight forward for our choice of 𝛿 and πœ– to see that𝐽12√(0,𝛿)<𝑛+1πœ‹ξ€œπ›Ώ02π‘‘πœƒ=πœ‹βˆšξ‚€π‘›+1arccos1βˆ’2log(𝑛𝐾)π‘›ξ‚βˆš=π‘œ(𝑛+1),(3.9)by condition (iii). As 𝐼1(βˆ’βˆž,∞)=𝐽1(0,𝛿)+𝐽1(𝛿,πœ‹/2), by (3.8) and (3.9) we see that𝐼1√(βˆ’βˆž,∞)βˆΌπ‘›+1.(3.10)Now we obtain an upper limit for 𝐼2 defined in (3.3). To this end, we let βˆšπ‘£=𝐾/(2𝐴). Then we have𝐼2||𝐾||(βˆ’βˆž,∞)β‰€βˆšξ€œ2πœ‹βˆžβˆ’βˆžπΆπ΄3ξ‚€βˆ’πΎexp22𝐴22𝑑π‘₯=βˆšπœ‹ξ€œβˆž0ξ‚€expβˆ’π‘£22π‘‘π‘£β‰€βˆšπœ‹.(3.11)This together with (3.10) proves that EN𝐾,π‘„βˆš(βˆ’βˆž,∞)βˆΌπ‘›+1. The theorem is proved for polynomial 𝑄(π‘₯) given in (1.3).

Let us now prove the theorem for polynomial 𝑃(π‘₯) given in (1.2), that isEN𝐾,π‘ƒβˆš(βˆ’βˆž,∞)βˆΌπ‘›+1.(3.12)The proof in this case repeats the proof for EN𝐾,𝑄(βˆ’βˆž,∞) 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 πœƒ to the original variable π‘₯ gives𝐽12(0,𝛿)=πœ‹ξ€œ0tan𝛿Δ𝐴2ξ‚€βˆ’π΅exp2𝐾22Ξ”22𝑑π‘₯<πœ‹ξ€œ0tan𝛿Δ𝐴2𝑑π‘₯,(3.13)where Ξ”/𝐴2 is given by (2.9). Then by the same reasoning as in the proof of Theorem 1.1,𝐽12√(0,𝛿)<𝑛+1πœ‹2√arctan(tan𝛿)=𝑛+1πœ‹π›Ώ=2βˆšπ‘›+1πœ‹ξ‚€arccos1βˆ’2log(𝑛𝐾)π‘›ξ‚ξ€·βˆš=π‘œξ€Έ,𝑛+1(3.14)by condition (iii). This completes the proof of Theorem 1.3.

4. Number of Maxima

In finding the expected number of maxima of 𝑃(π‘₯), we can find the expected number of zeros of its derivative π‘ƒξ…ž(π‘₯). To this end we first obtain the following characteristics needed in order to apply them into the Kac-Rice formula (2.1),𝐴2𝑀𝑃=varξ…žξ‚=(π‘₯)𝑛𝑗=0𝑛𝑗ξƒͺ(𝑗+1)π‘₯2𝑗=ξ‚€1+π‘₯2ξ‚π‘›βˆ’1ξ‚€1+π‘₯2+𝑛π‘₯2,𝐡(4.1)2𝑀=var(π‘ƒξ…žξ…žξ‚=(π‘₯)𝑛𝑗=0𝑛𝑗ξƒͺ𝑗2(𝑗+1)π‘₯2π‘—βˆ’2ξ‚€=𝑛1+π‘₯2ξ‚π‘›βˆ’3ξ‚€2+4𝑛π‘₯2+𝑛π‘₯4+𝑛2π‘₯4,𝐢(4.2)𝑀𝑃=covξ…ž(π‘₯),π‘ƒξ…žξ…žξ‚=(π‘₯)𝑛𝑗=0𝑛𝑗ξƒͺ𝑗(𝑗+1)π‘₯2π‘—βˆ’1ξ‚€=𝑛π‘₯1+π‘₯2ξ‚π‘›βˆ’2ξ‚€2+π‘₯2+𝑛π‘₯2.(4.3) Hence from (4.1)–(4.3) we obtainΞ”2𝑀=𝐴2𝑀𝐡2π‘€βˆ’πΆ2𝑀=𝑛1+π‘₯22π‘›βˆ’42+𝑛π‘₯4+𝑛2π‘₯4+2π‘₯2+2𝑛π‘₯2ξ‚„.(4.4)Now from (4.1) and (4.5) we haveΔ𝑀𝐴2𝑀=𝑛2+𝑛π‘₯4+𝑛2π‘₯4+2π‘₯2+4𝑛π‘₯21+π‘₯21+π‘₯2+𝑛π‘₯2.(4.5)As the value of π‘₯ increases, the dominating terms in (4.5) change. For accuracy therefore, the interval needs to be broken up. In this case, the interval (0,∞) was divided into two subintervals. First, choose πœ–<π‘₯<∞ such that πœ–=π‘›βˆ’1/4, thenΔ𝑀𝐴2π‘€βˆΌβˆšπ‘›1+π‘₯2.(4.6)Substituting into the Kac-Rice formula (2.1) yieldsEN𝑀𝑃1(πœ–,∞)βˆΌπœ‹ξ€œβˆžπœ–βˆšπ‘›1+π‘₯2βˆšπ‘‘π‘₯=𝑛2.(4.7)Now we choose 0<π‘₯<πœ–. Since for 𝑛 sufficiently large the term 𝑛2π‘₯4 is significantly larger than 𝑛π‘₯4 and also since for this range of π‘₯ we can see 2π‘₯2<1, we can obtain an upper limit for (4.5) asΔ𝑀𝐴2𝑀<𝑛3+2𝑛2π‘₯4+4𝑛π‘₯21+𝑛π‘₯2<𝑛3+6𝑛π‘₯2+3𝑛2π‘₯41+𝑛π‘₯2=√3𝑛.(4.8)Substituting this upper limit into Kac-Rice formula, we can seeENπ‘€π‘ƒξ€œ(0,πœ–)=πœ–0Ξ”π‘€πœ‹π΄2π‘€βˆšπ‘‘π‘₯<𝑛3π‘›πœ–=π‘œ1/4.(4.9)This together with (4.7) completes the proof of Theorem 1.4. To prove Corollary 1.5, it suffices to notice that since π‘„ξ…ž(π‘₯)=π‘ƒξ…ž(π‘₯) and π‘„ξ…žξ…ž(π‘₯)=π‘ƒξ…žξ…ž(π‘₯), all the arguments in the above proof apply to polynomial 𝑄(π‘₯), and we have therefore EN𝑀𝑃(π‘Ž,𝑏)=EN𝑀𝑄(π‘Ž,𝑏).


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