We consider private polynomial computation (PPC) over noncolluding coded
databases. In such a setting a user wishes to compute a multivariate polynomial
of degree at most $g$ over $f$ variables (or messages) stored in multiple
databases while revealing no information about the desired polynomial to the
databases. We construct two novel PPC schemes, where the first is a
generalization of our previous work in private linear computation for coded
databases. In this scheme we consider Reed-Solomon coded databases with
Lagrange encoding, which leverages ideas from recently proposed star-product
private information retrieval and Lagrange coded computation. The second scheme
considers the special case of coded databases with systematic Lagrange
encoding. Both schemes yield improved rates compared to the best known schemes
from the literature for a small number of messages, while in the asymptotic
case the rates match.