Alexander Levin
ABSTRACT
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.
- Cohn, R. M. Difference Algebra. Interscience, New York, 1965.Google Scholar
- A. Einstein. The Meaning of Relativity. Appendix II (Generalization of gravitation theory). Princeton University Press, Princeton, NJ, 1953, 153--165.Google Scholar
- Hrushovski, E. The Elementary Theory of the Frobenius Automorphisms. arXiv:math/0406514v1, 2004, 1--135. The updated version (2012): www.ma.huji.ac.il/ehud/FROB.pdf.Google Scholar
- E. R. Kolchin. The notion of dimension in the theory of algebraic differential equations. Bull. Amer. Math. Soc., 70 (1964), 570--573.Google ScholarCross Ref
- E. R. Kolchin. Differential Algebra and Algebraic Groups. Academic Press, 1973.Google Scholar
- M. V. Kondrateva, A. B. Levin, A. V. Mikhalev, and E. V. Pankratev. Differential and Difference Dimension Polynomials. Kluwer Acad. Publ., 1999.Google ScholarCross Ref
- A. B. Levin. Characteristic polynomials of filtered difference modules and of difference field extensions. Russian Math. Surveys, 33 (1978), no.3, 165--166.Google ScholarCross Ref
- A. B. Levin. Multivariable Difference Dimension Polynomials. Journal of Mathematical Sciences, 131, no. 6 (2005), 6060--6082.Google ScholarCross Ref
- A. B. Levin Computation of the Strength of Systems of Difference Equations via Generalized Gröbner Bases. Grobner Bases in Symbolic Analysis, Walter de Gruyter, 2007, 43--73.Google Scholar
- A. B. Levin. Gröbner bases with respect to several orderings and multivariable dimension polynomials. J. Symbolic Comput., 42 (2007), 561--578.Google ScholarDigital Library
- A. B. Levin Difference Algebra. Springer, 2008.Google Scholar
- A. B. Levin. Dimension Polynomials of Intermediate Fields of Inversive Difference Field Extensions. Lect. Notes in Comp. Sci., 9582 (2016), 362--376.Google ScholarDigital Library
- A. B. Levin and A. V. Mikhalev. Type and Dimension of Finitely Generated G-algebras. Contemporary Mathematics, 184 (1995), 275--280.Google Scholar
- Wibmer, M. Algebraic Difference Equations. Lecture Notes, University of Pennsylvania. http://www.mmrc.iss.ac.cn/mm2015/notes/wibmer1.pdfGoogle Scholar
Index Terms
- Reduction with Respect to the Effective Order and a New Type of Dimension Polynomials of Difference Modules
Recommendations
Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities
Let (p"n)"n be a sequence of orthogonal polynomials with respect to the measure @m. Let T be a linear operator acting in the linear space of polynomials P and satisfying deg(T(p))=deg(p)-1, for all polynomial p. We then construct a sequence of ...
Multivariate difference-differential dimension polynomials and new invariants of difference-differential field extensions
ISSAC '13: Proceedings of the 38th International Symposium on Symbolic and Algebraic ComputationWe introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of finitely generated difference-...
Generalized characteristic sets and new multivariate difference dimension polynomials
AbstractWe 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 . Using properties of ...
Comments