  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.
