Direct twisted Galois stratification

The theory ACFA admits a primitive recursive quantifier elimination procedure. It is therefore primitive recursively decidable

Twisted Galois stratification

We prove 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, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulae associated with finite Galois covers of difference schemes

Strongly étale difference algebras and Babbitt’s decomposition

We introduce a class of strongly etale difference algebras, whose role in the study of difference equations is analogous to the role of etale algebras in the study of algebraic equations. We deduce an improved version of Babbitt’s decomposition theorem and we present applications to difference algebraic groups and the compatibility problem