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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Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.
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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- Award ID(s):
- 1714425
- NSF-PAR ID:
- 10066959
- Date Published:
- Journal Name:
- Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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