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

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关于Weil下降多项式系统的最后一次下降度

给定多项式系统 $\mathcal{F}$在不一定为零维的有限域k上，我们考虑Weil下降${\mathcal{F}}^{\prime }$ 的 $\mathcal{F}$ 在子字段上 ${ķ}^{\prime }$。我们证明了一个定理，该定理与${\mathcal{F}}_{\mathrm{1个}}$ 和 ${\mathcal{F}}_{\mathrm{1个}}^{\prime }$，其中的零集 ${\mathcal{F}}_{\mathrm{1个}}$双射地对应于的k个理性点的集合$\mathcal{F}$和零集 ${\mathcal{F}}_{\mathrm{1个}}^{\prime }$ 是一组 ${ķ}^{\prime }$-Weil下降的理性点 ${\mathcal{F}}^{\prime }$。作为一个应用程序，我们得出最后一次跌落程度的上限${\mathcal{F}}_{\mathrm{1个}}^{\prime }$ 在这种情况下 $\mathcal{F}$ 是一组线性化多项式。