Abstract
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 \(\sigma\). 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 \(\sigma\) 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.
Similar content being viewed by others
References
Cohn, R.M.: Difference Algebra. Interscience, New York (1965)
Einstein, A.: The Meaning of Relativity. Appendix II (Generalization of gravitation theory), 4th ed. Princeton, 133–165
Kolchin, E.R.: The notion of dimension in the theory of algebraic differential equations. Bull Am. Math. Soc. 70, 570–573 (1964)
Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, Boston (1973)
Kondrateva, M. V., Levin, A. B., Mikhalev, A. V., Pankratev, E. V.: Differential and Difference Dimension Polynomials, Kluwer Acad. Publ (1998)
Levin, A.B.: Characteristic polynomials of filtered difference modules and difference field extensions. Russ. Math. Surv. 33(3), 165–166 (1978)
Levin, A.B.: Characteristic polynomials of inversive difference modules and some properties of inversive difference dimension. Russ. Math. Surv. 35(1), 217–218 (1980)
Levin, A. B.: Computation of the Strength of Systems of Difference Equations via Generalized Gröbner Bases. In: Gröbner Bases in Symbolic Analysis, Walter de Gruyter. Berlin–New York, pp. 43–73 (2007)
Levin, A.B.: Difference Algebra. Springer, New York (2008)
Levin, A.B.: Multivariate dimension polynomials of inversive difference field extensions. Lect. Notes Comput. Sci. 8372, 146–163 (2014)
Levin, A.B.: Multivariable difference dimension polynomials. J. Math. Sci. 131(6), 6060–6082 (2005)
Levin, A.B.: Groebner bases with respect to several orderings and multivariable dimension polynomials. J. Symb. Comput. 42, 561–578 (2007)
Levin, A.B.: A new type of difference dimension polynomials, Math. Comput. Sci., 16, no. 4, article 20, 13 pp (2022)
Levin, A.B.: Reduction with respect to the effective order and a new type of dimension polynomials of difference modules. In: Proceedings of ISSAC 2022. ACM Press, New York, 55–62 (2022)
Levin, A.B., Mikhalev, A.V.: Type and dimension of finitely generated G-algebras. Contemp. Math. 184, 275–280 (1995)
Mikhalev, A.V., Pankratev, E.V.: Differential dimension polynomial of a system of differential equations, Algebra (collection of papers), Moscow State Univ. Press, 57–67 (1980)
Acknowledgements
This research was supported by the NSF grant CCF–2139462.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Levin, A. Generalized characteristic sets and new multivariate difference dimension polynomials. AAECC 35, 31–53 (2024). https://doi.org/10.1007/s00200-023-00628-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-023-00628-0