# Zeros of random trigonometric polynomials

May 20, 2019 — October 14, 2019

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 squints 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. (Ben Hough et al. 2009; Pemantle and Rivin 2013; Krishnapur 2006)

I need more general results than i.i.d. coefficients; in particular, I need to relax the identical distribution assumption. 🚧TODO🚧 clarify

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