# Zeros of random trigonometric polynomials

For a certain nonconvex optimisation problem, I would like to know the expected number of real zeros of trigonometric polynomials

$0=\sum_{k=1}^{k=N}A(k)\sin(kx)B(k)\cos(kx)$

for given distributions over $$A(k)$$ and $$B(k)$$.

This is not exactly the usual sense of polynomial, although if one thinks about polynomials over the complex numbers and squint at it the relationship is not hard to see.

This problem is well studied for i.i.d. standard normal coefficients $$A(k),B(k)$$.

It turns out there are some determinantal point processes models for the distributions of zeros, which I should look into.

I need more general results than i.i.d. coefficients; in particular I need to relax the identical distribution assumption. 🏗

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