  We study expected values of the polynomials $P_N^{}(z)=\prod_{1\leq n\leq
N}(X_n^2+z^2)$ whose $2N$ zeros $\{\pm i X_k\}^{}_{k=1,...,N}$ are generated by
$N$ identically distributed multi-variate mean-zero normal random variables
$\{X_k\}^{N}_{k=1}$ with co-variance
${\rm{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma^2-1}{N})\delta_{k,l}+\frac{\sigma^2-1}{N}(1-\delta_{k,l})$.
In principle these can be evaluated in closed form for arbitrary $N$, yet
commonly available computer algebra handles only $N$ up to a dozen (due to
memory constraints). A list of the first three expected polynomials shows that
the expressions become unwieldy already for moderate $N$. On the other hand,
asymptotic evaluations of the large-$N$ regime for complex $z$ have
traditionally been limited to analytic expansion techniques, several rigorous
results are proved about this regime for complex $z$. Yet if $z$ is real one
can also compute the large-$N$ asymptotics in the "infinite-degree" limit with
the help of the familiar relative entropy principle for probability measures, a
rigorous proof of this fact is supplied. Computer algebra-generated evidence is
presented in support of a conjecture that a generalization of the relative
entropy principle to *{signed and complex measures}* governs the $N\to\infty$
asymptotics of the regime of imaginary $z$. Potential generalizations, in
particular to point vortex ensembles and the prescribed Gauss curvature
problem, and to random matrix ensembles, are emphasized.
