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

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...

The Zeros of a Random Polynomial

Correlations between Zeros of a Random Polynomial

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random p...

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 ...

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 levelu. In the derivation we assume chiefly that ξ(t), ξ(t), and ξ(t) 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 p...