  In this work, we consider private monomial computation (PMC) for replicated
noncolluding databases. In PMC, a user wishes to privately retrieve an
arbitrary multivariate monomial from a candidate set of monomials in $f$
messages over a finite field $\mathbb F_q$, where $q=p^k$ is a power of a prime
$p$ and $k \ge 1$, replicated over $n$ databases. We derive the PMC capacity
under a technical condition on $p$ and for asymptotically large $q$. The
condition on $p$ is satisfied, e.g., for large enough $p$. Also, we present a
novel PMC scheme for arbitrary $q$ that is capacity-achieving in the asymptotic
case above. Moreover, we present formulas for the entropy of a multivariate
monomial and for a set of monomials in uniformly distributed random variables
over a finite field, which are used in the derivation of the capacity
expression.
