New Multivariate Dimension Polynomials of Inversive Difference Field Extensions

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial. Then we develop the corresponding method of characteristic sets and apply it to prove the existence and obtain a method of computation of multivariate dimension polynomials of a new type that describe the transcendence degrees of intermediate fields of finitely generated inversive difference field extensions obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of $\sigma$ are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than standard (univariate) difference dimension polynomials. We also show how the obtained results can be applied to the equivalence problem for systems of algebraic difference equations.Comment: arXiv admin note: text overlap with arXiv:1207.4757, arXiv:1302.150

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

Geometric Difference Galois Theory

Die vorliegende Arbeit entwickelt eine Galoistheorie für Differenzengleichungen basierend auf differenzenalgebraischer Geometrie. Hierbei wird ein System von gewöhnlichen Differenzengleichungen durch eine Differenzenerweiterung beschrieben, und die Galoisgruppen sind Gruppenschemata vom endlichen Typ über den Konstanten. Die Galoisgruppen müssen weder linear noch reduziert sein. Das Hauptresultat ist eine Charakterisierung jener Differenzenerweiterungen, die eine gutartige Galoistheorie zulassen, durch eine Normalitätseigenschaft. Inspiration für diesen Zugang war die Arbeit von J. Kovacic über die Galoistheorie von stark normalen Differentialerweiterungen

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

Hilbert Irreducibility above algberaic groups

The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.Comment: 22 page

Contributions to the model theory of partial differential fields

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the existence and properties of the model companion of the theory of partial differential fields with an automorphism. The approach taken here to these subjects is to relativize the algebro geometric notions of prolongation and D-variety to differential notions with respect to a fixed differential structure.Comment: PhD Thesi