Twisted Galois stratification (Q639610)

This paper develops the theory of twisted Galois stratification in order to describe first-order definable sets in the language of difference rings over algebraic closures of finite fields equipped with powers of the Frobenius automorphism. After introducing basic concepts of the theory of difference schemes and their morphisms, as well as the notions of a (normal) Galois stratification \(\mathcal{A}\) on a difference scheme (\(X, \sigma\)) and the Galois formula associated with \(\mathcal{A}\), the author develops difference algebraic geometry (in particular, the theory of generalized difference schemes). The main result of the paper is a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence of this theorem, the author obtains an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes. In addition, the paper presents a number of new results on the category of difference schemes, Babbitt's decomposition, and effective difference algebraic geometry.