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

   https://doi.org/10.1145/3476446.3535497  

   Levin, Alexander (July 2022, 2022 International Symposium on Symbolic and Algebraic Computation)

   General Chair: Marc Moreno Chair: Lihong Zhi (Ed.)

   We introduce a new type of reduction in a free difference module over a difference field that uses a generalization of the concept of effective order of a difference polynomial. Then we define the concept of a generalized characteristic set of such a module, establish some properties of these characteristic sets and use them to prove the existence, outline a method of computation and find invariants of a dimension polynomial in two variables associated with a finitely generated difference module. As a consequence of these results, we obtain a new type of bivariate dimension polynomials of finitely generated difference field extensions. We also explain the relationship between these dimension polynomials and the concept of Einstein’s strength of a system of difference equations. 

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2. Dimension Quasi-polynomials of Inversive Difference Field Extensions with Weighted Translations

   Levin, A. (November 2017, Mathematical Aspects of Computer and Information Sciences)

   We consider Hilbert-type functions associated with finitely generated inversive difference field extensions and systems of algebraic difference equations in the case when the translations are assigned positive integer weights. We prove that such functions are quasi-polynomials that can be represented as alternating sums of Ehrhart quasi-polynomials of rational conic polytopes. In particular, we generalize the author's results on difference dimension polynomials and their invariants to the case of inversive difference fields with weighted basic automorphisms. 

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3. Generalized characteristic sets and new multivariate difference dimension polynomials

   https://doi.org/10.1007/s00200-023-00628-0  

   Levin, Alexander (October 2023, Applicable Algebra in Engineering, Communication and Computing)

   We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations σ. Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of σ 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 previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial. 

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4. Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals. The Case of One Translation

   Levin, A. (July 2018, Proceedings of the 43rd International Symposium on Symbolic and Algebraic Computation (ISSAC 2018))

   We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. As a consequence, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. We also discuss applications of our results to systems of algebraic difference-differential equations. 

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5. Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.

   Levin, A. (June 2018, Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018)

   Multivariate dimension polynomials associated with finitely generated differential and difference field extensions arise as natural generalizations of the univariate differential and difference dimension polynomials. It turns out, however, that they carry more information about the corresponding extensions than their univariate counterparts. We extend the known results on multivariate dimension polynomials to the case of difference-differential field extensions with arbitrary partitions of sets of basic operators. We also describe some properties of multivariate dimension polynomials and their invariants. 

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