Random Algebraic Polynomials Research Articles

Real almost zeros of random polynomials with complex coefficients

We present a simple formula for the expected number of times that a complex‐valued Gaussian stochastic process has a zero imaginary part and the absolute value of its real part is bounded by a constant value M. We show that only some mild conditions on the stochastic process are needed for our formula to remain valid. We further apply this formula to a random algebraic polynomial with complex coefficients. We show how the above expected value in the case of random algebraic polynomials varies for different behaviour of M.

Zeros of an algebraic polynomial with nonequal means randomcoefficients

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0 + a1x + a2x2 + ⋯+an − 1xn−1. The coefficients aj(j = 0, 1, 2, …, n − 1) are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval (−1, 1) and the mean of the last group in (−∞, −1)∪(1, ∞).

On the Expected Number of Real Zeros of Certain Gaussian Random Polynomials

Let be a random algebraic polynomial where the coefficients A 0,A 1,… form a sequence of centered Gaussian random variables. Moreover, assume that the increments Δ j =A j −A j−1,j=0,1,2,… are independent, A −1=0. The coefficients A 0, A 1, … A n can be considered as n consecutive observations of a Brownian motion. We provide an explicit formula for the expected density of the number of real zeros of Q n (x). We observe that the expected density of real zeros on (−1/p,1/p) has the limit p/(1−p 2 x 2), n→∞ where p 2j =Var(Δ j ), and we obtain the asymptotic behaviour of the expected number of real zeros for the case that p=1, which exhibits new features in the study of random algebraic polynomials.

Real zeros of classes of random algebraic polynomials

There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0 + a1x + a2x2 + ⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients it is shown that the above expected number is asymptotic to O(logn). This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non‐identically distributed such that the variance of the jth term is the expected number of zeros of the polynomial increases to . The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients { a j } n−1 j=0 of the polynomial T( x)= a 0+ a 1 x+ a 2 x 2+⋯+ a n−1 x n−1 are normally distributed, with mean E( a j )= μ j+1 , where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of | μ|<1 and | μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.

Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients

In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known.

Zeros of random hyperbolic and random algebraic polynomials

Zeros of random hyperbolic and random algebraic polynomials

Extrema of random algebraic polynomials with non-identically distributed normal coefficients

AbstractAn asymptotic estimate is derived for the expected number of extrema of a polynomial whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form a 0 ( n − 1 0 ) 1 / 2 + a 1 ( n − 1 1 ) 1 / 2 x + a 2 ( n − 1 2 ) 1 / 2 x 2 + ⋯ + a n − 1 ( n − 1 n − 1 ) 1 / 2 x n − 1 a_0\binom {n-1}{0}^{1/2}+a_1\binom {n-1}{1}^{1/2}x +a_2\binom {n-1}{2}^{1/2}x^2+\cdots +a_{n-1}\binom {n-1}{n-1}^{1/2}x^{n-1} where a j , j = 0 , 1 , 2 , … , n − 1 a_{j}, j= 0, 1, 2, \ldots , n-1 , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the x x axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form a 0 + a 1 x + a 2 x 2 + ⋯ + a n − 1 x n − 1 a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1} which was previously the most studied.

Local maxima of a random algebraic polynomial

We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t), ξ′(t), and have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to above is then used to find the expected number of maxima below the level u for the random algebraic polynomial. This result highlights the very pronounced difference in the behaviour of the random algebraic polynomial on the interval (−1, 1) compared with the intervals (−∞, −1) and (1, ∞). It is also shown that the number of maxima below the zero level is no longer O(logn) on the intervals (−∞, −1) and (1, ∞).

Strong result for real zeros of random algebraic polynomials

An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non‐identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Expected Number of Real Zeros of Lévy and Harmonizable Stable Random Polynomials

Abstract Assuming (A 0, A 1, . . . , An ) is a jointly symmetric α-stable random vector, 0 &lt; α ≤ 2, a general formula for the expected number of real zeros of the random algebraic polynomial was obtained by Rezakhah in 1997. Rezakhah's formula is applied to the case where (A 0, . . . , An ) is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived for the expected number of real zeros of .

Complex Roots of a Class of Random Algebraic Polynomials

The average density of the distribution of complex roots of equation η0g0+η1g1z+η2g2z2+···+ηn−1gn−1zn−1=Kwith random coefficients ηjand forz∈C,gj≡1,j=0,1,…,n−1, andKa complex constant with equal real and imaginary parts is known. The present paper provides a formula for such an average density for any constant realgjand ηj=aj+bj, where theajs andbjs,j=0,1,…,n−1, are real standard normal independent random variables, andK=K1+iK2, whereK1andK2are constants independent ofz. The limiting behaviour of this density function asntends to infinity for several special selectedgjs is obtained.

The average number of point of inflection of random algebraic polynomials

Let be a sequence of random variables normally distributed with mean zero and variance one. There are many known results for the expected number of real zeros and number of maxima (minima) of the polynomial for large n The present paper provides the asymptotic behavior of the average number of points of inflection of such polynomial

On the expected number of zero crossings of a random algebraic polynomial

On the expected number of zero crossings of a random algebraic polynomial

Complex Roots of a Random Algebraic Polynomial

This paper, for any constantK, provides an exact formula for the average density of the distribution of the complex roots of equation η0+η1z+η2z2+···+ηn−1zn−1=Kwhere ηj=aj+ibjand {aj}n−1j=0and {bj}n−1j=0are sequences of independent identically and normally distributed random variables andKis a complex number withKas its real and imaginary parts. The case of real roots of the above equation with real coefficients andK,z∈Ris well known. Further we obtain the limiting behaviour of this distribution function asntends to infinity.

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial \(a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} \), where aj (j=0, 1,...,n−1) are independent, normally distributed random variables with mean μ and variance one. It is shown that for nonzero μ, the expected number of maxima is asymptotic to \(((\sqrt 3 + 1)/4\pi )\)log n, when n is large.

Real Zeros of Random Algebraic Polynomials

Real Zeros of Random Algebraic Polynomials

A lower bound for the variance of real zeros of random polynomials

Let be a random id algebraic polynomial where {ai} is a sequence of independent identically distributed ( iid ) standard normal random variables. In this paper we have obtained a lower bound for the variance of the number of real zeros .of the random algebraic polynomials Qn(x). We have shown that the bound is for sufficiently large n. Our estimate is times that of Maslova (1974). We have also presented a graph and a comparison table illustrating the values of variances

The average number of level crossings of a random algebraic polynomial

Let be a sequence of normally distributed independent random variables with mathematical expectation μ≠0, and variance unity. Here for n→∞ we find the asymptotic relation for the mathematical expectation of the number of real roots of the equation where K is any non zero real constant, permitted to be a function of degree n. It is shown that the results are valid even for K→∞ as long as