The complexity of solving Weil restriction systems

Abstract: 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 $\mathcal{F}$.