Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

Abstract EN

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the form y = P n t = n 1 a 1 cosh t + n 2 a 2 cosh 2 t + ⋯ + n n a n cosh n t , where a 1 , … , a n is a sequence of standard Gaussian random variables, is n / 2 + o p ( 1 ) .

It is shown that the asymptotic value of expected number of times the polynomial crosses the level y = K is also n / 2 as long as K does not exceed 2 n e μ ( n ) , where μ ( n ) = o ( n ) .

The number of oscillations of P n ( t ) about y = K will be less than n / 2 asymptotically only if K = 2 n e μ ( n ) , where μ ( n ) = O ( n ) or n - 1 μ ( n ) → ∞ .

In the former case the number of oscillations continues to be a fraction of n and decreases with the increase in value of μ ( n ) .

In the latter case, the number of oscillations reduces to o p ( n ) and almost no trace of the curve is expected to be present above the level y = K if μ ( n ) / ( n log n ) → ∞ .