On real zeros of random polynomials with hyperbolic elements

Abstract

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomial g1coshx+2g2cosh2x+…+ngncoshnx where gj,(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently large n this asymptotic value is (1/π)logn.

Keywords

• Number of real zeros
• Kac-Rice formula
• random algebraic polynomial
• random trigonometric polynomial.