Symmetric polynomials are polynomials that are invariant under the action of the
symmetric group, and they play an integral role in mathematics. The space of quasiin-
variant polynomials, polynomials that are invariant under the action of the symmetric
group to a certain order, were introduced by Feigin and Veselov. These spaces are
modules over the ring of symmetric polynomials, and their Hilbert series in elds of
characteristic 0 were also computed by Feigin and Veselov.
In this paper, we study the Hilbert series of these spaces in elds of positive char-
acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted
quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the
space is twisted by a monomial. We extend some of their results to spaces twisted by
a product of smooth functions and compute the Hilbert series of the space in certain
cases.