Dimension Polynomials of Intermediate Fields of Inversive Difference Field Extensions

Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions

“…, t p+q ) satisfies conditions (ii) of Theorem 6. Statement (iii) of Theorem 6 can be obtained in the same way as statement (iii) of Theorem 2 of [13]. )…”

“…Proof. In order to show the existence and uniqueness of the desired mapping µ U , one can just mimic the proof of the corresponding statement for chains of prime differential ideals given in [3, Section 1] (see also [11,Proposition 4.1] and [13,Section 4] where similar arguments were applied to differential and inversive difference field extensions, respectively). Namely, let us set µ U (F, E) = −1 if F = E or the field extension F/E is algebraic.…”

“…Note that we consider arbitrary (not necessarily inversive) difference-differential extensions of an inversive difference-differential field. In the particular case of purely differential extensions and in the case of inversive difference field extensions, the existence and properties of dimension polynomials were obtained in [11] and [13]. The main problem one runs into while working with a non-inversive difference (or difference-differential) field extension is that the translations are not invertible and there is no natural difference (respectively, difference-differential) structure on the associated module of Kähler differentials.…”

“…Dimension polynomials associated with finitely generated differential field extensions were introduced by E. Kolchin in [4]; their properties and various applications can be found in his fundamental monograph [5,Chapter 2]. A similar technique for difference and inversive difference field extensions was developed in [7], [8], [12], [13] and some other works of the author. Almost all known results on differential and difference dimension polynomials can be found in [6] and [10].…”

Reduction with Respect to the Effective Order and a New Type of Dimension Polynomials of Difference Modules

“…field extension of 𝐾 generated by the images of 𝑦 𝑖 in 𝑅/𝑃. The dimension polynomial of this extension, therefore, characterizes the ideal 𝑃; assigning such polynomials to prime reflexive difference polynomial ideals has led to a number of new results on dimension of difference varieties (see [3] and [14]) and on the Krull-type dimension of difference algebras (see [13], [6,Section 7.2], and [11,Section 4.6]) and difference field extensions (see [12]). Another important application of difference dimension polynomials is based on the fact that the univariate difference dimension polynomial of a system of algebraic difference equations (defined as the dimension polynomial of the difference field extension associated with the system) expresses the A. Einstein's strength of this system (see [9] and [11,Chapter 7]).…”