Given a polynomial system $\mathcal{ F}$ over a finite field $k$ which is not necessarily of dimension zero, we consider the Weil descent $\mathcal{ F}'$ of $\mathcal{ F}$ over a subfield $k'$. We prove a theorem which relates the last fall degrees of $\mathcal{ F}_1$ and $\mathcal{ F}'_1$, where the zero set of $\mathcal{ F}_1$ corresponds bijectively to the set of $k$-rational points of $\mathcal{ F}$, and the zero set of $\mathcal{ F}'_1$ is the set of $k'$-rational points of the Weil descent $\mathcal{ F}'$. As an application we derive upper bounds on the last fall degree of $\mathcal{ F}'_1$ in the case where $\mathcal{ F}$ is a set of linearized polynomials.
Categories: math.AG math.NT