Zeros of an algebraic polynomial with nonequal means random coefficients

Zeros of an algebraic polynomial with nonequal means random coefficients

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 aremainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are...

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a0+a1x+a2x+ · · · + an−1xn−1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π) logn. For the dependent cases studied so far it is shown that this asymptotic value remains O(logn). In ...

The Real Zeros of a Random Polynomial with Dependent Coefficients

Abstract. Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of 2 π logn, as n → ∞. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case ...

On Algebraic Polynomials with Random Coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form a0 (n−1 0 )1/2 + a1 (n−1 1 )1/2 x + a2 (n−1 2 )1/2 x2 + · · · + an−1 (n−1 n−1 )1/2 xn−1 where aj , j = 0, 1, 2, . . . , 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...

The Zeros of a Random Polynomial