Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The density of complex zeros of random sums   Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically
distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j =
0}^{N}$ be a sequence of given analytic functions that are real-valued on the
real number line. We prove an exact formula for the expected density of the
distribution of complex zeros of the random equation $\sum_{j = 0}^{N} \eta_{j}
f_{j} (z) = \mathbf{K}$, where $\mathbf{K} \in \mathds{C}$. The method of proof
employs a formula for the expected absolute value of quadratic forms of
Gaussian random variables. We then obtain the limiting behaviour of the density
function as $N$ tends to infinity and provide numerical computations for the
density function and empirical distributions for random sums with certain
functions $f_{j} (z)$. Finally, we study the case when the $f_{j} (z)$ are
polynomials orthogonal on the real line and the unit circle.

34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg