  In this paper, we study the number of real roots of random trigonometric
polynomials with iid coefficients. When the coefficients have zero mean, unit
variance and some finite high moments, we show that the variance of the number
of real roots is asymptotically linear in terms of the expectation;
furthermore, the multiplicative constant in this linear relationship depends
only on the kurtosis of the common distribution of the polynomial's
coefficients. This result is in sharp contrast to the classical Kac polynomials
whose corresponding variance depends only on the first two moments. Our result
is perhaps the first paper to establish the variance for general distribution
of the coefficients including discrete ones, for a model of random polynomials
outside the family of the Kac polynomials. Our method gives a fine comparison
framework throughout Edgeworth expansion, asymptotic Kac-Rice formula and a
detailed analysis of characteristic functions.
