Mean number of real zeros of a random trigonometric polynomial

Mean Number of Real Zeros of a Random Hyperbolic Polynomial

Consider the random hyperbolic polynomial, f(x) = 1a1 coshx+···+np × an coshnx, in which n and p are integers such that n ≥ 2, p ≥ 0, and the coefficients ak(k = 1,2, . . . ,n) are independent, standard normally distributed random variables. If νnp is the mean number of real zeros of f(x), then we prove that νnp = π−1 logn+ O{(logn)1/2}.

Covariance of the number of real zeros of a random trigonometric polynomial

For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ) = ∑n j=0{aj cos jθ + bj sin jθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/ √ 3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value f...

How many zeros of a random polynomial are real?

Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, . . . , tn) projected onto the surface of the unit sphere, divided by π. The probability density...

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 ...

The Zeros of a Random Polynomial