Introduction and preliminaries.If (K, ∂) is a differentially closed field of characteristic 0, an algebraic D-variety over K is an algebraic variety over K together with an extension of the derivation ∂ to a derivation of the structure sheaf of X. Welet D denote the category of algebraic D-varieties and G the full subcategory whose objects are algebraic D-groups. D and G were essentially introduced by Buium and G was exhaustively studied in [2]. The category D is closely related but not identical to the class of sets of finite Morley rank definable in the structure (K,+, ・, ∂) (which is essentially the class of “finitedimensional differential algebraic varieties”). However an object ofD can also be considered as a first order structure in its own right by adjoining predicates for algebraic D-subvarieties of Cartesian powers. In a similar fashion D can be considered as a many-sorted first order structure. In [5] it was shown that the many-sorted “reduct” G has quantifier-elimination. We point out in this paper an easy example showing:

PROPOSITION 1.1. D does not have quantifier-elimination.

The notion of a “complete variety” in algebraic geometry is fundamental. Over C these are precisely the varieties which are compact as complex spaces. Kolchin [4] introduced completeness in the context of differential algebraic geometry. This was continued by Pong [12] who obtained some interesting results and examples. We will give a natural definition of completeness for algebraic D-varieties. The exact relationship to the notion studied by Kolchin and Pong is unclear: in particular I do not know whether any of the examples of new “complete” ∂-closed sets in [12] correspond to complete algebraic D varieties in our sense.

Introduction.This paper is to some extent a continuation of my introductive and programmatic paper [Ku3]. In that paper I pointed out that the ramification theoretical defect of finite extensions of valued fields is responsible for the problems we have when we deal with the model theory of valued fields, or try to prove local uniformization in positive characteristic.

In the present paper I will discuss the connection between the defect and additive polynomials. I will state and prove basic facts about additive polynomials and then treat several instances where they enter the theory of valued fields in an essential way that is particularly interesting for model theorists and algebraic geometers. I will show that non-commutative structures (skew polynomial rings) play an essential role in the structure theory of valued fields in positive characteristic. Further, I will state the main open questions. I will also include some exercises.

In the next section, I will give an introduction to additive polynomials and describe the reasons for their importance in the model theory of valued fields. For the convenience of the reader, I outline the characterizations of additive polynomials in Section 3 and the basic properties of rings of additive polynomials in Section 4. For more information on additive polynomials, the reader may consult [Go], cf. also [O1], [O2], [Wh1], [Wh2], [Ku4], [Dr–Ku]. The remaining sections of this paper will then be devoted to the proofs of some of the main theorems stated in Section 2.

Introduction. Propositional calculi are usually defined by systems of axioms schemes and rules of inference. Natural problems arising in general study of logical calculi, for example, the problemof equivalence or the problem of determining for arbitrary calculus whether it is consistent or not, are, in general, undecidable. The first undecidability results for propositional calculi were found by S. Linial and E. Post in 1949 [16]. In particular, it was proved that the property “to be an axiomatization of the classical propositional logic” is undecidable. In 1963 A. V. Kuznetsov [13] found an essential extension of this result. He proved that for every superintuitionistic logic L the property “to be an axiomatization of L” is undecidable. In particular, consistency andmany other properties of calculi are, in general, undecidable. In [3] A. V. Chagrov has given a survey of results on undecidable properties of propositional calculi. Here we concentrate on decidable properties.

When we restrict ourself by considering particular families of calculi, for instance, propositional calculi extending intuitionistic or some modal logic, many important properties of calculi appear to be decidable (see a survey in [4]). When the rules of inference are fixed, for any given finite system of additional axioms, one can effectively decide consistency problem for normal modal logics, tabularity and interpolation problems for extensions of the intuitionistic logic or of the modal system S4 and some other problems.