On the last fall degree of Weil descent polynomial systems

Abstract

Given a polynomial system F over a finite field k which is not necessarily of dimension zero, we consider the Weil descent F′ of F over a subfield k′. We prove a theorem which relates the last fall degrees of F1 and F1′, where the zero set of F1 corresponds bijectively to the set of k-rational points of F, and the zero set of F1′ is the set of k′-rational points of the Weil descent F′. As an application we derive upper bounds on the last fall degree of F1′ in the case where F is a set of linearized polynomials.