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Given a polynomial system $\mathcal{F}$ over a finite field $k$ which is notnecessarily of dimension zero, we consider the Weil descent $\mathcal{F}'$ of$\mathcal{F}$ over a subfield $k'$. We prove a theorem which relates the lastfall 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 $k'$-rationalpoints of the Weil descent $\mathcal{F}'$. As an application we derive upperbounds on the last fall degree of $\mathcal{F}'_1$ in the case where$\mathcal{F}$ is a set of linearized polynomials.

On the last fall degree of Weil descent polynomial systems