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:多变量多元分子方程式的解析度为通过 Groebner 基础方法计算系统解决方案的复杂度提供了一个上限。 在本文中, 我们考虑通过 el 限制 calars 获得的多元货币系统。 后者是一种算术构造, 鉴于Galois 字段的有限扩展 $k\ hookrightrow K$, 与一个系统 $\ mathcal{F} 的关联值, 定义在 $\ mathrm{Weil} (\ mathcal{F}) 上方, 定义在 $\ mathcal{F} 上方, 定义在 $\\ mathrm} (mathcal{F} ) 上方块中, 以 美元和 $\ mathrm{ Weil} 通过 Weil 限制 获得的多元货币系统 的复杂度。