The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner bases methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension $k\hookrightarrow K$, associates to a system $\mathcal{F}$ defined over $K$ a system $\mathrm{Weil}(\mathcal{F})$ defined over $k$, in such a way that the solutions of $\mathcal{F}$ over $K$ and those of $\mathrm{Weil}(\mathcal{F})$ over $k$ are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system $\mathrm{Weil}(\mathcal{F})$ obtained via Weil restriction in terms of algebraic invariants of the system $\mathcal{F}$.

翻译：多元多项式方程组的求解度给出了通过Gröbner基方法计算系统解复杂度的上界。本文考虑通过Weil标量限制得到的多项式系统。后者是一种算术构造，给定有限伽罗瓦域扩张$k\hookrightarrow K$，它将定义在$K$上的系统$\mathcal{F}$关联到定义在$k$上的系统$\mathrm{Weil}(\mathcal{F})$，使得$\mathcal{F}$在$K$上的解与$\mathrm{Weil}(\mathcal{F})$在$k$上的解之间存在自然双射。本文利用系统$\mathcal{F}$的代数不变量，给出了通过Weil限制得到的多项式系统$\mathrm{Weil}(\mathcal{F})$求解复杂度的上界。