  We pursue various restricted variable generalizations of the
Chevalley-Warning theorem for low degree polynomial systems over a finite
field. Our first such result involves variables restricted to Cartesian
products of the Vandermonde subsets of $\F_q$ defined by G\'acs-Weiner and
Sziklai-Tak\'ats. We then define an invariant $\uomega(X)$ of a nonempty subset
of $\F_q^n$. Our second result involves $X$-restricted variables when the
degrees of the polynomials are small compared to $\uomega(X)$. We end by
exploring various classes of subsets for which $\uomega(X)$ can be bounded from
below.
