Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

Abstract

The expected number of real zeros of an algebraic polynomial 𝑎_𝑜+𝑎₁𝑥+𝑎₂𝑥²+⋯+𝑎_𝑛𝑥^𝑛 with random coefficient 𝑎_𝑗,𝑗=0,1,2,…,𝑛 is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the 𝑗th coefficient is var(𝑎_𝑗)=^𝑛𝑗. It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume 𝐸(𝑎_𝑗)=^𝑛𝑗𝜇^𝑗+1 and var(𝑎_𝑗)=^𝑛𝑗𝜎^2𝑗. We show how the above expected number of real zeros is dependent on values of 𝜎² and 𝜇 in various cases.

1. Origin of Polynomials

Let (Ω,Pr,𝒜) be a fixed probability space, and for 𝜔∈Ω let {𝑎_𝑗(𝜔)}^𝑛𝑗=0 be a sequence of independent identically distributed random variables defined on Ω. There has been considerable work on obtaining the expected number of real zeros of algebraic𝑃_𝑛(𝑥,𝜔)≡𝑃_𝑛(𝑥)=^𝑛_𝑗=0𝑎_𝑗(𝜔)𝑥^𝑗,(1.1)and trigonometric ∑^𝑛𝑗=0𝑎_𝑗(𝜔)cos𝑗𝜃 polynomials with random coefficients 𝑎_𝑗(𝜔)s. The study of the random algebraic polynomials was initiated by Kac [1], and the recent works include [2, 3]. It is shown that under general assumptions for the distribution of coefficients the expected number of real zeros is asymptotic to (2/𝜋)log𝑛 as 𝑛→∞. For the case of random trigonometric polynomials, Dunnage [4] obtained the first result which was later generalized by Wilkins Jr. [5, 6] and recently studied in [7, 8]. It is shown that, again for a wide class of distributions for the coefficients, there are significantly more real zeros in the case of trigonometric polynomial compared with the algebraic case. The asymptotic value for the expected number of zeros for the latter case is 2𝑛/√3. Besides the comprehensive book of Bharucha-Reid and Sambandham [9], the earlier results of general topics on random polynomials are reviewed in [10].

Motivated by the interesting work of Edelman and Kostlan [11], who, among others, considered polynomials of the form ∑^𝑛𝑗=0𝑎_𝑗(𝜔)(^𝑛𝑗)^1/2𝑥^𝑗, [2, 12] obtained many characteristics, like the number of real zeros or the number of maxima of these types of polynomials. This is interesting as they showed that for this case of nonidentically distributed coefficients the expected number of real zeros is 𝑂(√𝑛), which is significantly more than the classical algebraic case but less than that of trigonometric polynomials. Also in this direction of nonidentical coefficients, a case in which the mean of coefficients 𝑎_𝑗(𝜔) increases with 𝑗 is studied in [3, 13]. Now it would be interesting to study a random polynomial formed by combining the above two distribution laws. It is natural to ask, for instance, what would be the behavior of 𝑃_𝑛(𝑥) in (1.1) if for constants 𝜇 and 𝜎 the mean and variance of coefficients are 𝐸(𝑎_𝑗(𝜔))=(^𝑛𝑗)𝜇^𝑗+1 and var(𝑎_𝑗(𝜔))=(^𝑛𝑗)𝜎^2𝑗.

With the latter assumption of the distribution of the coefficients, we first show that if 𝜇=0, the expected number of real zeros of 𝑃_𝑛(𝑥) denoted by 𝐸𝑁_𝑛,𝑃(0,∞)≡𝐸𝑁_𝑛(0,∞) is independent of 𝜎. The case of nonzero 𝜇 is studied in Theorem 1.2. The analysis for the general case is complicated, and we only give the result for a case that 𝜇=𝜎². We prove the following theorem.

Theorem 1.1. For 𝜇=0 and 𝜎²>0, the expected number of real zeros of 𝑃_𝑛(𝑥) is independent of 𝜎². That is 𝐸𝑁_𝑛(−∞,0)=𝐸𝑁_𝑛(0,∞)=√𝑛2.(1.2)

The analysis for the case of 𝜇≠0 would be complicated. Without loss of much generality and certainly interest, we restrict ourselves to the case of 𝜇=𝜎². We prove the following theorem.

Theorem 1.2. The expected number of real zeros of 𝑃_𝑛(𝑥) for different values of 𝜇 satisfies 𝐸𝑁_𝑛(0,∞)⎧⎪⎨⎪⎩∼=𝑂√(1)𝑛,if𝜇=𝜎>1,21−arctan2√𝜇1−𝜇,if0<𝜇=𝜎<1.(1.3)For 𝑥 negative and for every 𝜇=𝜎², 𝐸𝑁_𝑛(−∞,0)∼√𝑛2.(1.4)

2. Moments

In order to obtain the expected number of real zeros we use a generalization of the well-known Kac-Rice formula initiated in [1, 14, 15]. To this end, we need the following moments of 𝑃_𝑛(𝑥) and its dervative 𝑃^𝑛(𝑥). First, we assume the general assumptions on the means and the variances of coefficients as stated above. That is, 𝐸(𝑎_𝑗)=(^𝑛𝑗)𝜇^𝑗+1 and var(𝑎_𝑗)=(^𝑛𝑗)𝜎^2𝑗. Since these coefficients are independent, it is easy to show𝛼=𝐸(𝑃_𝑛(𝑥))=𝜇^𝑛_𝑗=0𝑛𝑗(𝑥𝜇)^𝑗𝛽=𝐸(𝑃_𝑛=𝜇(1+𝜇𝑥)^𝑛,(2.1)(𝑥))=𝜇^2𝑛_𝑗=0𝐴𝑗𝑛𝑗(𝜇𝑥)^𝑗−1=𝑛𝜇²(1+𝜇𝑥)^𝑛−1,(2.2)₂=var(𝑃_𝑛(𝑥))=^𝑛_𝑗=0𝑛𝑗(𝜎𝑥)^2𝑗𝐵=(1+𝜎²𝑥²)^𝑛,(2.3)₂=var(𝑃^𝑛(𝑥))=𝜎^2𝑛_𝑗=0𝑗²𝑛𝑗(𝜎𝑥)²=𝑛𝜎²(1+𝜎²𝑥²)^𝑛−2(1+𝑛𝜎²𝑥²),(2.4) and finally𝐶=cov(𝑃_𝑛(𝑥),𝑃^𝑛(𝑥))=𝜎²𝑥^𝑛_𝑗=0𝑗𝑛𝑗(𝜎𝑥)^2𝑗−2=𝑛𝜎²𝑥(1+𝜎²𝑥²)^𝑛−1.(2.5) Then from (2.3)–(2.5) we can obtainΔ²=𝐴²𝐵²−𝐶²=𝑛𝜎²(1+𝜎²𝑥²)^2𝑛−2.(2.6)With the above notations, we can now write the Kac-Rice for the expected number of real zeros of 𝑃_𝑛(𝑥) in the interval (𝑎,𝑏) as, see also [10, page 43],𝐸𝑁_𝑛(𝑎,𝑏)=𝐼₁(𝑎,𝑏)+𝐼₂(𝑎,𝑏),(2.7)where𝐼₁(𝑎,𝑏)=^𝑏𝑎Δ𝜋𝐴²exp−𝛼²𝐵²+𝛽²𝐴²−2𝛼𝛽𝐶𝐼2Δ²𝑑𝑥,(2.8)₂(𝑎,𝑏)=^𝑏𝑎√2|𝛽𝐴²−𝐶𝛼|𝜋𝐴³exp−𝛼²2𝐴²erf|𝛽𝐴²−𝐶𝛼|√2𝐴Δ𝑑𝑥,(2.9) where as usual erf(𝑥)=∫^𝑥0exp(−𝑡²)𝑑𝑡. Now we can progress and evaluate further the following terms required in the Kac-Rice formulae (2.7)–(2.9). From (2.1)–(2.5) we can derive

=𝑛𝜇₂𝛼²𝐵²+𝛽²𝐴²−2𝛼𝛽𝐶(1+𝜇𝑥)^2𝑛−2(1+𝜎²𝑥²)^𝑛−2(𝑛𝜎⁴𝑥²+𝜎²+2𝜇𝑥𝜎²+𝜇²𝑥²𝜎²+𝑛𝜇²−2𝑛𝜇𝜎²𝑥).(2.10)This together with (2.6) yields𝛼²𝐵²+𝛽²𝐴²−2𝛼𝛽𝐶2Δ²=𝜇²(1+𝜇𝑥)^2𝑛−2{𝑛(𝜎²𝑥−𝜇)²+𝜎²(1+𝜇𝑥)²}2𝜎²(1+𝜎²𝑥²)^𝑛.(2.11)

3. Proof of Theorems

First in the case of 𝜇=0 from (2.7) and (2.3)–(2.6) by letting 𝑦=𝜎𝑥 we can show𝐸𝑁_𝑛(0,∞)=√𝑛𝜋^∞0𝜎1+𝜎²𝑥²=√𝑑𝑥𝑛𝜋^∞0𝑑𝑦1+𝑦²=√𝑛2.(3.1)This proves Theorem 1.1. Now we proceed with the more general case of 𝜇≠0. As explained above, in order to simplify the analysis we let 𝜇=𝜎². This yields (2.11) to𝛼²𝐵²+𝛽²𝐴²−2𝛼𝛽𝐶2Δ²=𝑓_𝑛(𝑥,𝜇)𝑔_𝑛(𝑥,𝜇),(3.2)where𝑔_𝑛(𝑥,𝜇)=𝑛𝜇²(𝑥−1)²+𝜇(1+𝜇𝑥)²,(3.3)and for all sufficiently large 𝑛,𝑓_𝑛(𝑥,𝜇)=(1+𝜇𝑥)^2𝑛−2(1+𝜇𝑥²)^𝑛∼(1+𝜇𝑥)^2𝑛(1+𝜇𝑥²)^𝑛=1+𝜇²𝑥²+2𝜇𝑥1+𝜇𝑥²^𝑛.(3.4)Now we assume 𝑥>0. Then if we let 𝜇>1, since1+𝜇²𝑥²+2𝜇𝑥1+𝜇𝑥²>1,(3.5)we can see that 𝑓_𝑛(𝑥,𝜇)→∞ as 𝑛→∞. Therefore the exponential term that appears in (2.8) tends to zero exponentially fast. Hence the only contribution to 𝐸𝑁_𝑛(0,∞) is from 𝐼₂(0,∞). In the following, we show that the latter is 𝑂(1). To this end, we note that since from the definition for all 𝑥, erf(𝑥)≤√𝜋/2, then𝐼₂(0,∞)<1√2𝜋^∞0𝛽𝐴²−𝐶𝛼𝐴³exp−𝛼²2𝐴²𝑑𝑥.(3.6)Now we let 𝑢=𝛼/𝐴, and since (𝑑/𝑑𝑥)(𝛼/𝐴)=(𝛽𝐴²−𝛼𝐶)/𝐴³ from (3.6) we obtain𝐼₂(0,∞)<1√2𝜋^∞0exp−𝑢²2𝑑𝑢≤21.(3.7)This completes the first part of Theorem 1.2. On the other hand, if 𝜇<1, the behavior of 𝑓_𝑛(𝑥,𝜇) will depend on 𝑥. That is for 0<𝑥<2/(1−𝜇), 𝑓_𝑛(𝑥,𝜇)→∞ as 𝑛→∞ and for 𝑥>2/(1−𝜇), 𝑓_𝑛(𝑥,𝜇)→0 as 𝑛→∞. Therefore the only contribution to 𝐸𝑁_𝑛(0,∞) from 𝐼₁ is in the interval (2/(1−𝜇),∞) as 𝐼₁(0,2/(1−𝜇)) will tend to zero exponentially fast. Also for 𝜈=√𝜇𝑥,𝐼₁21−𝜇,∞∼√𝑛𝜋^{∞2/(1−𝜇)}√𝜇1+𝜇𝑥²𝑑𝑥=√𝑛𝜋^{∞2/(1−𝜇)}𝑑𝜈1+𝜈²∼√𝑛21−arctan2√𝜇1−𝜇.(3.8)The above argument for 𝐼₂(0,∞) in (3.7) remains valid, and therefore we have proof of the first part of Theorem 1.2.

For 𝑥<0 without loss of generality, we only consider the case of 𝜇>0 (since 𝜇=𝜎²). For this case 𝑔_𝑛(𝑥,𝜇) remains positive. However, for 𝑥²>𝜖/𝜇, where for 𝑎=1−loglog𝑛¹⁰/log𝑛 we let 𝜖=𝑛^−𝑎, we have, (see also [10, page 31]),(1+𝜇𝑥²)^𝑛>(1+𝜖)^𝑛{(1+𝑛^−𝑎)^𝑛𝑎}^𝑛1−𝑎=exp(𝑛^1−𝑎)∼𝑛¹⁰.(3.9)Hence𝑓_𝑛(𝑥,𝜇)<(1+𝜇𝑥²)^−𝑛<𝑛⁻¹⁰,(3.10)which tends to zero very fast as 𝑛→∞. Therefore the exponential term in 𝐼₁ tends to be one, and hence𝐼₁𝜖𝜇,∞∼𝑛𝜋^∞√_𝜖/𝜇𝑑𝑥1+𝑥²∼√𝑛2.(3.11)Also in the interval (0,√𝜖/𝜇),𝐼₁0,𝜖𝜇<^{√0𝜖/𝜇}Δ𝜋𝐴²𝑑𝑥<√𝑛𝜇𝜋^{√0𝜖/𝜇}𝑑𝑥1+𝜇𝑥²∼√𝑛2𝜋arctan𝜖,(3.12)which is small. This completes the proof of Theorem 1.2.