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Specializations over difference fields

3C is termed a monadic extension of W. For the sake of brevity, we use the term pathological to designate an extension which either is incompatible with some other extension of ff or is a monadic extension of W. The existence of pathological extensions has implications for both the abstract and the analytic theory of difference equations [5]. The fundamental importance of incompatible extensions, for example, becomes clear when one examines the body of theorems in difference algebra concerning specializations. Indeed the standard theorems of algebraic geometry regarding specializations go over to difference algebra only if one imposes conditions precluding incompatibility [6]. The existence of monadic extensions on the other hand, threatens seriously to complicate the task of constructing a Galois theory for difference fields. Thus the significance of pathological extensions to difference algebra is unquestionably great, but our knowledge of these extensions is quite limited. Cohn [5] has shown that a difference field admits finitely generated pathological extensions only if it admits stuch extensions of order zero, i.e. algebraic over the ground field. Hence fields which are algebraically closed admit no finitely generated pathological extensions. These are the only general results regarding finitely generated pathological extensions yet obtained. In view of Cohn's results it is natural to inquire if the existence of finitely generated pathological extensions of order zero in turn implies the existence of pathological extensions which are of finite degree over the ground field. Our primary purpose in this paper is to investigate this supposition; our principal result is that this supposition is true. In ?2 we prove a decomposition theorem for normal extensions of order

/pdf/finitely-generated-pathological-extensions-of-difference-49cohunkif.pdf

Finitely generated pathological extensions of difference fields

of algebraic fields, that is, fields in the ordinary sense, onily if the characteristic p exceed 0. In this case an extension which is monadic in the sense just described may be produced by adjoining a p-th root not already in the field. Our purpose in this paper is to present in organized form the still rudimentary theory of these phenomena, and to point out their decisive importance in the algebraic theory of difference equations. They seem, in fact, to mark a point byond which one can no longer use the theory of polynomial ideals or the algebraic theory of differential equations as a guide to the study of difference equations, but must expect phenomena which are sti generis. Our results also suggest interesting questions concerning field structure. It would, for example, be desirable to characterize those fields which possess incompatible or monadic extensions. Our principal result is a first step toward such a characterization; fields which are algebraically closed have no finite extensions of these types.

Extensions of difference fields.

Joseph Fels Ritt

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.

Dimension polynomials in the generalized difference case

Complete Difference Ideals

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Complete Difference Ideals

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Specializations over difference fields

Finitely generated pathological extensions of difference fields

Extensions of difference fields.

Joseph Fels Ritt

Dimension polynomials in the generalized difference case

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