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Zeros of random trigonometric polynomials

For a certain nonconvex optimisation problem, I need 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 too hard to see.

This problem is well studied for i.i.d. standard normal coefficients \(A(k),B(k)\). I need more general results; in particular I need to relax the identical distribution assumption.

It turns out there is some kind of point process result here, maybe impinging upon determinantal point processes.

Reading: [@FlascheExpected2017;@GarciaBrief2002;@VanderbeiComplex2015;@DumitrescuPositive2017;@EdelmanHow1995].

Refs