On Different Classes of Algebraic Polynomials with Random Coefficients

Abstract

The expected number of real zeros of the polynomial of the form 𝑎₀+𝑎₁𝑥+𝑎₂𝑥²+⋯+𝑎_𝑛𝑥^𝑛, where 𝑎₀,𝑎₁,𝑎₂,…,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)log𝑛. In this paper, we show that this asymptotic value increases significantly to √𝑛+1 when we consider a polynomial in the form 𝑎₀^𝑛0^1/2𝑥/√1+𝑎₁^𝑛1^1/2𝑥²/√2+𝑎₂^𝑛2^1/2𝑥³/√3+⋯+𝑎_𝑛^𝑛𝑛^1/2𝑥^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.

1. Introduction

The classical random algebraic polynomial has previously been defined as𝑇(𝑥)≡𝑇_𝑛(𝑥,𝜔)=^𝑛_𝑗=0𝑎_𝑗(𝜔)𝑥^𝑗,(1.1)where, for (Ω,𝒜,Pr) a fixed probability space, {𝑎_𝑗(𝜔)}^𝑛𝑗=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)log𝑛. 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 (^𝑛𝑗). 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 (^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(𝑥)≡𝑃_𝑛(𝑥,𝜔)=^𝑛_𝑗=0𝑎_𝑗𝑛𝑗^1/2𝑥^𝑗+1√𝑗+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(𝑥)≡𝑄_𝑛(𝑥,𝜔)=^𝑛_𝑗=0𝑎_𝑗𝑛𝑗^1/2𝑥^𝑗+1√𝑗+1+√𝑛𝑎^∗+1,(1.3)where 𝑎^∗ is mutually independent of and has the same distribution as {𝑎_𝑗}^𝑛𝑗=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 𝐸𝑁_0,𝑃(−∞,∞)∼√𝑛+1.(1.4)

Corollary 1.2. With the same assumption as Theorem 1.1 for the coefficients 𝑎_𝑗 and 𝑎^∗ one has 𝐸𝑁_0,𝑄(−∞,∞)=√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≤𝑒^𝑛𝑛²(ii)1,𝑛²(iii)𝐾₂=𝑜𝐾²,=𝑜𝑒^𝑛𝑛².(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 𝐸𝑁_𝐾,𝑄(−∞,∞)∼𝐸𝑁_𝐾,𝑃(−∞,∞)∼√𝑛+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 EN^𝑀𝑃 for polynomial 𝑃(𝑥) given in (1.2) and, since the event of tangency at the 𝑥-axis 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 𝐸𝑁^𝑀𝑃(−∞,∞)∼√𝑛.(1.8)

Corollary 1.5. With the above assumptions for the coefficients 𝑎_𝑗 and 𝑎^∗ one has 𝐸𝑁^𝑀𝑄(−∞,∞)∼√𝑛.(1.9)

2. Proof of Theorem 1.1

We use a well-known Kac-Rice formula, [1, 9], in which it is proved thatEN_0,𝑃(𝑎,𝑏)=^𝑏𝑎Δ𝜋𝐴²𝑑𝑥,(2.1)where 𝑃^(𝑥) represents the derivative with respect to 𝑥 of 𝑃(𝑥). We denote𝐴²=var𝑃(𝑥),𝐵²=var𝑃^(𝑥),𝐶=cov𝑃(𝑥),𝑃^(𝑥),Δ²=𝐴²𝐵²−𝐶².(2.2)Now, with our assumptions on the distribution of the coefficients, it is easy to see that𝐴²=^𝑛_𝑗=0𝑛𝑗𝑥𝑗+1^2𝑗+2=1+𝑥²^𝑛+1𝐵𝑛+1−𝑛+11,(2.3)₂=^𝑛_𝑗=0𝑛𝑗(𝑗+1)𝑥^2𝑗𝐶=_𝑛=1+𝑥²^𝑛−11+𝑥²+𝑛𝑥²,(2.4)_𝑗=0𝑛𝑗𝑥^2𝑗+1=𝑥1+𝑥²^𝑛.(2.5) We note that, for all sufficiently large 𝑛 and 𝑥 bounded away from zero, from (2.3) we have𝐴²∼1+𝑥²^𝑛+1𝑛+1.(2.6)This together with (2.1), (2.4), and (2.5) yieldsEN₀(−∞,∞)∼2𝜋^𝜖0Δ𝐴²𝑑𝑥+2𝜋^∞𝜖√𝑛+11+𝑥²𝑑𝑥,(2.7)where 𝜖>0, 𝜖→0 as 𝑛→∞. The second integral can be expressed as2√𝑛+1𝜋𝜋2−arctan𝜖→√𝑛+1as𝑛→∞.(2.8)In the first integral, the expression (Δ/𝐴²) has a singularity at 𝑥=0:Δ𝐴²=⎷(𝑛+1)1+𝑥²^2𝑛−1+𝑥²^𝑛−11+𝑥²+𝑛𝑥²1+𝑥²^𝑛+1−1².(2.9)Notice that the expression in (2.9) is bounded from above:Δ𝐴²<√(𝑛+1)(1−𝐷)1+𝑥²,(2.10)where𝐷=1+𝑛𝑥²1+𝑥²^𝑛−1−1+𝑥²^𝑛1+𝑥²^𝑛−1²=(𝑛−1)1+𝑥²^𝑛−2+(𝑛−2)1+𝑥²^𝑛−3+⋯+31+𝑥²²+21+𝑥²+11+𝑥²^𝑛−1+1+𝑥²^𝑛−2+⋯+1+𝑥²+1².(2.11) When 𝑥=0, we have𝐷=𝑛²−𝑛2𝑛²(2.12)and therefore Δ𝐴²<𝑛+1√2𝑛∼𝑛+12,(2.13)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 that1>𝐷>∑^{𝑛−2𝑗=0}(1+𝑗)𝑛²1+𝑥²^2𝑛−2>0,(2.14)and thereforeΔ𝐴²<√𝑛+11+𝑥².(2.15)Hence, the first integral that appears in (2.7) is bounded from above as follows:2𝜋^𝜖0Δ𝐴²𝑑𝑥<2𝜋^𝜖0√𝑛+11+𝑥²𝑑𝑥=2(arctan𝜖)√𝑛+1𝜋=𝑜√𝑛+1(2.16)by the choice of 𝜖. 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(−∞,∞)∼√𝑛+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 𝐵² and 𝐶. However, for 𝐴² we can obtain the exact value rather than the asymptotic value. To this end, we can easily see that𝐴^2𝑄=var𝑄(𝑥)=^𝑛_𝑗=0𝑛𝑗𝑥𝑗+1^2𝑗+2+𝑛+11=1+𝑥²^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:EN_0,𝑄(−∞,∞)=1𝜋^∞0√𝑛+11+𝑥²𝑑𝑥=√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 coefficientsEN_𝐾(𝑎,𝑏)=𝐼₁(𝑎,𝑏)+𝐼₂(𝑎,𝑏)(3.1)with 𝐼₁(𝑎,𝑏)=^𝑏𝑎Δ𝜋𝐴²exp−𝐵²𝐾²𝐼2Δ²𝑑𝑥,(3.2)₂(𝑎,𝑏)=^𝑏𝑎√2𝐾𝐶𝜋𝐴³exp−𝐾²2𝐴²erf−√𝐾2𝐴𝐶Δ𝑑𝑥,(3.3) where, as usual, erf(𝑥)=∫^𝑥0exp(−𝑡)𝑑𝑡≤√𝜋/2. Since changing 𝑥 to −𝑥 leaves the distribution of the coefficients unchanged, EN_𝐾(−∞,0)=EN_𝐾(0,∞). Hence to what follows we are only concerned with 𝑥≥0. Using (2.3)–(2.5) and (3.2) we obtain𝐼₁(−∞,∞)=2√𝑛+1𝜋^∞011+𝑥²exp−𝐾²(𝑛+1)1+𝑥²+𝑛𝑥²21+𝑥²^𝑛+1𝑑𝑥.(3.4)Using substitution 𝑥=tan𝜃 in (3.4) we can see that𝐼₁(−∞,∞)=𝐽₁0,𝜋2=2√𝑛+1𝜋^0𝜋/2exp−𝐾²(𝑛+1)21+𝑛sin²𝜃cos^2𝑛𝜃𝑑𝜃,(3.5)where the notation 𝐽₁ 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/(𝑛𝜖)), where 𝜖=1/{2log(𝑛𝐾)}. This choice of 𝜖 is indeed possible by condition (i). Now since cos𝜃<(1−1/(𝑛𝜖)), we can show thatcos^2𝑛𝜃<1−1𝑛𝜖^2𝑛=1−1𝑛𝜖^−𝑛𝜖^−2/𝜖∼exp−2𝜖→0(3.6)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 𝜃𝐾²𝑛²cos^2𝑛−2𝜃<𝐾²𝑛²exp−2𝜖=𝐾²𝑛²exp−4log(𝑛𝐾)=(𝐾𝑛)⁻²→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 𝜃 and we can easily see by the bounded convergence theorem and condition (iii) that𝐽₁𝛿,𝜋2∼√𝑛+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 𝜖 to see that𝐽₁(0,𝛿)<2√𝑛+1𝜋^𝛿0𝑑𝜃=2𝜋√𝑛+1arccos1−2log(𝑛𝐾)𝑛=𝑜(√𝑛+1),(3.9)by condition (iii). As 𝐼₁(−∞,∞)=𝐽₁(0,𝛿)+𝐽₁(𝛿,𝜋/2), by (3.8) and (3.9) we see that𝐼₁(−∞,∞)∼√𝑛+1.(3.10)Now we obtain an upper limit for 𝐼₂ defined in (3.3). To this end, we let 𝑣=𝐾/(√2𝐴). Then we have𝐼₂(−∞,∞)≤||𝐾||√2𝜋^∞−∞𝐶𝐴³exp−𝐾²2𝐴²𝑑𝑥=√2𝜋^∞0exp−𝑣²𝑑𝑣≤2√𝜋.(3.11)This together with (3.10) proves that EN_𝐾,𝑄(−∞,∞)∼√𝑛+1. The theorem is proved for polynomial 𝑄(𝑥) given in (1.3).

Let us now prove the theorem for polynomial 𝑃(𝑥) given in (1.2), that isEN_𝐾,𝑃(−∞,∞)∼√𝑛+1.(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 gives𝐽₁(0,𝛿)=2𝜋^0tan𝛿Δ𝐴²exp−𝐵²𝐾²2Δ²𝑑𝑥<𝜋2^0tan𝛿Δ𝐴²𝑑𝑥,(3.13)where Δ/𝐴² is given by (2.9). Then by the same reasoning as in the proof of Theorem 1.1,𝐽₁(0,𝛿)<2√𝑛+1𝜋arctan(tan𝛿)=2√𝑛+1=2√𝜋𝛿𝑛+1𝜋arccos1−2log𝑛(𝑛𝐾)=𝑜√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(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),𝐴^2𝑀=var𝑃^(𝑥)=^𝑛_𝑗=0𝑛𝑗(𝑗+1)𝑥^2𝑗𝐵=1+𝑥²^𝑛−11+𝑥²+𝑛𝑥²,(4.1)_2𝑀=var(𝑃^(𝑥)=^𝑛_𝑗=0𝑛𝑗𝑗²(𝑗+1)𝑥^2𝑗−2𝐶=𝑛1+𝑥²^𝑛−32+4𝑛𝑥²+𝑛𝑥⁴+𝑛²𝑥⁴,(4.2)_𝑀=cov𝑃^(𝑥),𝑃^(𝑥)=^𝑛_𝑗=0𝑛𝑗𝑗(𝑗+1)𝑥^2𝑗−1=𝑛𝑥1+𝑥²^𝑛−22+𝑥²+𝑛𝑥².(4.3) Hence from (4.1)–(4.3) we obtainΔ^2𝑀=𝐴^2𝑀𝐵^2𝑀−𝐶^2𝑀=𝑛1+𝑥²^2𝑛−42+𝑛𝑥⁴+𝑛²𝑥⁴+2𝑥²+2𝑛𝑥².(4.4)Now from (4.1) and (4.5) we haveΔ_𝑀𝐴^2𝑀=𝑛2+𝑛𝑥⁴+𝑛²𝑥⁴+2𝑥²+4𝑛𝑥²1+𝑥²1+𝑥²+𝑛𝑥².(4.5)As the value of 𝑥 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 𝜖<𝑥<∞ such that 𝜖=𝑛^−1/4, thenΔ_𝑀𝐴^2𝑀∼√𝑛1+𝑥².(4.6)Substituting into the Kac-Rice formula (2.1) yieldsEN^𝑀𝑃(𝜖,∞)∼1𝜋^∞𝜖√𝑛1+𝑥²𝑑𝑥=√𝑛2.(4.7)Now we choose 0<𝑥<𝜖. Since for 𝑛 sufficiently large the term 𝑛²𝑥⁴ is significantly larger than 𝑛𝑥⁴ and also since for this range of 𝑥 we can see 2𝑥²<1, we can obtain an upper limit for (4.5) asΔ_𝑀𝐴^2𝑀<𝑛3+2𝑛²𝑥⁴+4𝑛𝑥²1+𝑛𝑥²<𝑛3+6𝑛𝑥²+3𝑛²𝑥⁴1+𝑛𝑥²=√3𝑛.(4.8)Substituting this upper limit into Kac-Rice formula, we can seeEN^𝑀𝑃(0,𝜖)=^𝜖0Δ_𝑀𝜋𝐴^2𝑀𝑑𝑥<√3𝑛𝜖=𝑜𝑛^1/4.(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 𝑄^(𝑥)=𝑃^(𝑥) and 𝑄^(𝑥)=𝑃^(𝑥), all the arguments in the above proof apply to polynomial 𝑄(𝑥), and we have therefore EN^𝑀𝑃(𝑎,𝑏)=EN^𝑀𝑄(𝑎,𝑏).