Restricted Variable Chevalley-Warning Theorems

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 ${\text{F}}_q$ defined by G'acs-Weiner and Sziklai-Tak'ats. We then define an invariant $\text{uomega}(X)$ of a nonempty subset of ${\text{F}}_q^n$. Our second result involves $X$-restricted variables when the degrees of the polynomials are small compared to $\text{uomega}(X)$. We end by exploring various classes of subsets for which $\text{uomega}(X)$ can be bounded from below.