Abstract
In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference–differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchberger-type algorithm for constructing relative Gröbner bases of filtered free modules.
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Buchberger, B.: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Ph.D. Thesis, University of Innsbruck (1965)
Dönch, C., Winkler, F.: Bivariate difference-differential polynomials and their computation in Maple. In: Proceedings of 8th International Conference on Applied Informatics, pp. 211–218. Eger, Hungary (2010)
Dönch, C.: Standard bases in finitely generated difference-skew-differential modules and their application to dimension polynomials. Ph.D. Thesis, Research Institute for Symbolic Computation (RISC), Linz, Austria (2012)
Dönch, C.: Characterization of relative Gröbner Bases. J. Symb. Comput. 55(1), 19–29 (2013)
Dönch, C., Levin, A.: Bivariate dimension polynomials and new invariants of finitely generated D-modules. Internat. J. Algebra Comput. 23(7), 1625–1651 (2013)
Fürst, C.: Axiomatic description of Gröbner reduction. Ph.D. Thesis, Research Institute for Symbolic Computation (RISC), Linz, Austria (2016)
Fürst, C., Landsmann, G.: Computation of dimension in filtered free modules by Gröbner reduction. In: Proceedings of ISSAC, Bath, UK, pp. 181–188 (2015)
Fürst, C., Landsmann, G.: Three examples of Gröbner reduction over noncommutative rings. RISC report series 15–16, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria (2016)
Futorny, V., Hartwig, J.: Multiparameter twisted Weyl algebras. J. Algebra 357, 69–93 (2012)
Huang, G., Zhou, M.: Termination of algorithm for computing relative Gröbner bases and difference differential dimension polynomials. Front. Math. China 10(3), 635–648 (2015)
Kandri-Rodi, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput. 9(1), 1–26 (1990)
Kondrateva, M., Levin, A., Mikhalev, A., Pankratev, E.: Differential and Difference Dimension Polynomials. Kluwer Academic Publishers, Massachusetts (1998)
Levin, A.: Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials. J. Symb. Comput. 30(4), 357–382 (2000)
Levin, A.: Characteristic polynomials of finitely generated modules over Weyl algebras. Bull. Austral. Math. Soc. 61, 387–403 (2000)
Levin, A.: Gröbner bases with respect to several term orderings and multivariate dimension polynomials. In: Proceedings of ISSAC, pp. 251–260. Waterloo, Ontario, Canada (2007)
Levin, A.: Gröbner bases with respect to several orderings and multivariable dimension polynomials. J. Symb. Comput. 42(5), 561–578 (2007)
Levin, A.: Difference Algebra. Springer, New York (2008)
Oaku, T., Shimoyama, T.: A Gröbner basis method for modules over rings of differential operators. J. Symb. Comput. 18(3), 223–248 (1994)
Pusz, W., Woronowicz, S.: Twisted second quantization. Rep. Math. Phys. 27(2), 231–257 (1989)
Winkler, F., Zhou, M.: Gröbner bases in difference–differential modules. Proc. ISSAC. ACM Press, New York (2006)
Winkler, F., Zhou, M.: Gröbner bases in difference–differential modules and difference–differential dimension polynomials. Sci. China. Ser. A Math. 51(9), 1732–1752 (2008)
Winkler, F., Zhou, M.: Computing difference–differential dimension polynomials by relative Gröbner bases in difference–differential modules. J. Symb. Comput. 43(10), 726–745 (2008)
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Fürst, C., Levin, A. Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules. Math.Comput.Sci. 11, 329–339 (2017). https://doi.org/10.1007/s11786-017-0317-1
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DOI: https://doi.org/10.1007/s11786-017-0317-1