Zeros of an algebraic polynomial with nonequal means random coefficients

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial ao a1x a2x · · · anx with random coefficient aj , 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 aj ( n j ) . It is sh...

Real Zeros of Random Algebraic Polynomials with Binomial Elements

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0 + a1x + a2x + ···+ an−1xn−1. The coefficients aj ( j = 0,1,2, . . . ,n− 1) are assumed to be independent normal random variables withmean zero. For integers m and k = O(logn)2 the variances of the coefficients are assumed to have nonidentical value var(aj) = ( k−1 j−ik ) , where ...

Expected Number of Local Maxima of Some Gaussian Random Polnomials

Expected Discrepancy for Zeros of Random Algebraic Polynomials

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of t...

On the average number of sharp crossings of certain Gaussian random polynomials