Research Article | Open Access
K. Farahmand, A. Grigorash, B. McGuinness, "On Different Classes of Algebraic Polynomials with Random Coefficients", International Journal of Stochastic Analysis, vol. 2008, Article ID 189675, 8 pages, 2008. https://doi.org/10.1155/2008/189675
On Different Classes of Algebraic Polynomials with Random Coefficients
The expected number of real zeros of the polynomial of the form , where is a sequence of standard Gaussian random variables, is known. For large it is shown that this expected number in is asymptotic to . In this paper, we show that this asymptotic value increases significantly to when we consider a polynomial in the form 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.
The classical random algebraic polynomial has previously been defined aswhere, for a fixed probability space, is a sequence of independent random variables defined on . For large, the expected number of real zeros of , in the interval , defined by , is known to be asymptotic to . For this case the coefficients are assumed to be identical normal standard. This asymptotic value was first obtained by the pioneer work of Kac  and was recently significantly improved by Wilkins , who reduced the error term involved in this asymptotic formula to . Since then, many other mathematical properties of have been studied and they are listed in  and more recently in .
The other class of random polynomials is introduced in an interesting article of Edelman and Kostlan  in which the th coefficients of in (1.1) have nonidentical variance . It is interesting to note that in this case the expected number of zeros significantly increased to , 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 , maximized at the middle term of 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 , which together with its mathematical interest motivated us to study the polynomialAs 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 definewhere is mutually independent of and has the same distribution as . We prove the following.
Theorem 1.1. When the coefficients of 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 and one has
Also of interest is the expected number of times that a curve representing the polynomial cuts a level . We assume is any constant such thatFor example, any absolute constant satisfies these conditions. Defining as the expected number of real roots of , we can generalize the above theorem to the following one.
Theorem 1.3. When the coefficients have the same distribution as in Theorem 1.1, and 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 for polynomial given in (1.2) and, since the event of tangency at the -axis has probability zero, we note that this is asymptotically the same as the expected number of real zeros of . In the following theorem, we give the expected number of maxima of the polynomial.
Theorem 1.4. With the above assumptions on the coefficients , then the asymptotic estimate for the expected number of maxima of is
Corollary 1.5. With the above assumptions for the coefficients and one has
2. Proof of Theorem 1.1
We use a well-known Kac-Rice formula, [1, 9], in which it is proved thatwhere represents the derivative with respect to of . We denoteNow, with our assumptions on the distribution of the coefficients, it is easy to see that We note that, for all sufficiently large and bounded away from zero, from (2.3) we haveThis together with (2.1), (2.4), and (2.5) yieldswhere , as . The second integral can be expressed asIn the first integral, the expression has a singularity at :Notice that the expression in (2.9) is bounded from above:where When , we haveand therefore which means that the integrand in the first integral of (2.7) is bounded for every . When , it can easily be seen thatand thereforeHence, 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 and . However, for we can obtain the exact value rather than the asymptotic value. To this end, we can easily see thatSubstituting 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:This gives the proof of Corollary 1.2.
3. Level Crossings
To find the expected number of -level crossings, we use the following extension to the Kac-Rice formula as it was used in . It is shown that in the case of normal standard distribution of the coefficientswith where, as usual, Since changing to leaves the distribution of the coefficients unchanged, . Hence to what follows we are only concerned with . Using (2.3)–(2.5) and (3.2) we obtainUsing substitution in (3.4) we can see thatwhere the notation emphasizes integration in . In order to progress with the calculation of the integral appearing in (3.5), we first assume , where where . This choice of is indeed possible by condition (i). Now since we can show thatas . 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 by condition (ii). Therefore, for all sufficiently large , the argument of the exponential function in (3.5) is reduced to zero, and hence the integrand is not a function of and we can easily see by the bounded convergence theorem and condition (iii) thatSince the argument of the exponential function appearing in (3.5) is always negative, it is straight forward for our choice of and to see thatby condition (iii). As , by (3.8) and (3.9) we see thatNow we obtain an upper limit for defined in (3.3). To this end, we let . Then we haveThis together with (3.10) proves that . The theorem is proved for polynomial given in (1.3).
Let us now prove the theorem for polynomial given in (1.2), that isThe proof in this case repeats the proof for 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 to the original variable giveswhere is given by (2.9). Then by the same reasoning as in the proof of Theorem 1.1,by condition (iii). This completes the proof of Theorem 1.3.
4. Number of Maxima
In finding the expected number of maxima of , we can find the expected number of zeros of its derivative . To this end we first obtain the following characteristics needed in order to apply them into the Kac-Rice formula (2.1), Hence from (4.1)–(4.3) we obtainNow from (4.1) and (4.5) we haveAs the value of increases, the dominating terms in (4.5) change. For accuracy therefore, the interval needs to be broken up. In this case, the interval was divided into two subintervals. First, choose such that , thenSubstituting into the Kac-Rice formula (2.1) yieldsNow we choose . Since for sufficiently large the term is significantly larger than and also since for this range of we can see we can obtain an upper limit for (4.5) asSubstituting this upper limit into Kac-Rice formula, we can seeThis together with (4.7) completes the proof of Theorem 1.4. To prove Corollary 1.5, it suffices to notice that since and , all the arguments in the above proof apply to polynomial , and we have therefore .
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