Toward involutive bases over effective rings

Springer Science and Business Media LLC - Tập 31 - Trang 359-387 - 2020
Michela Ceria1, Teo Mora2
1Department of Computer Science, University of Milan, Milano, Italy
2Department of Mathematics, University of Genoa, Genova, Italy

Tóm tắt

In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases.

Tài liệu tham khảo

Apel, J.: Gröbnerbasen in Nichetkommutativen Algebren und ihre Anwendung. Dissertation, Leipzig (1988) Apel, J.: Computational ideal theory in finitely generated extension rings. Theor. Comput. Sci. 224, 1–33 (2000) Bertone, C., Cioffi, F., Lella, P., Roggero, M.: Upgraded methods for the effective computation of marked schemes on a strongly stable ideal. J. Symb. Comput. 50, 263–290 (2013) Bertone, C., Lella, P., Roggero, M.: A Borel open cover of the Hilbert scheme. J. Symb. Comput. 53, 119–135 (2013) Bertone, C., Cioffi, F., Roggero, M.: Macaulay-like marked bases. J. Algebra Its Appl. 16(05), 1750100 (2017) Bertone, C., Cioffi, F., Roggero, M.: Double-generic initial ideal and Hilbert scheme. Annali di Matematica Pura ed Applicata 196(1), 19–41 (2017) Bertone, C., Cioffi, F., Roggero, M.: Smoothable Gorenstein points via marked schemes and double-generic initial ideals. Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1592034 Buchberger, B.: A criterion for detecting unnecessary reduction in the construction of Gröbner bases. L. N. Comp. Sci. 72, 3–21 (1979) Bueso, J., Gomez-Torrecillas, J., Verschoren, A.: Methods in Non-commutative Algebra. Kluwer, Dordrecht (2003) Ceria, M., Mora, T.: Buchberger–Zacharias theory of multivariate ore extensions. J. Pure Appl. Algebra 221(12), 2974–3026 (2017) Ceria, M., Mora, T., Roggero, M.: Term-ordering free involutive bases. J. Symb. Comput. 68, 87–108 (2015) Ceria, M.: Combinatorial decompositions for monomial ideals, submitted Ceria, M., Mora, T.: Buchberger-Weispfenning theory for effective associative rings. J. Symb. 83, 112–146 (2017) Cioffi, F., Roggero, M.: Flat families by strongly stable ideals and a generalization of Gröbner bases. J. Symb. Comput. 46(9), 1070–1084 (2011) Donato, L., Traverso, C.: Experimenting the Gröbner basis algorithm with AlP I system. In: Proceedings ISSAC ’89, pp. 192–198. ACM (1989) Gebauer, R., Möller, H.M.: On an installation of Buchbgerger’s algorithm. J. Symb. Comput. 6, 275–286 (1988) Gebauer, R., Möller, H.M.: Buchberger’s algorithm and staggered linear bases. In: Proceedings of SYMSAC’86. ACM, pp. 218–221(1986) Gerdt, V.P., Blinkov, Y.A.: Involutive bases of Polynomial Ideals. Math. Comput. Simul. 45, 543–560 (1998) Gerdt, V.P., Blinkov, Y.A.: Minimal involutive bases. Math. Comput. Simul. 45, 519–541 (1998) Gerdt, V.P., Blinkov, Y.A.: Janet-like monomial division. In: International Workshop on Computer Algebra in Scientific Computing. Springer, Berlin (2005) Gerdt, V.P., Blinkov, Y.A.: Janet-like Groebner bases. In: International Workshop on Computer Algebra in Scientific Computing. Springer, Berlin (2005) Gerdt, V., Blinkov, Y., Yanovich, D.: Construction of Janet Bases I. Monomial Bases. In: Computer Algebra in Scientific Computing CASC, pp. 233–247 (2001) Gunther, N.: Sur les modules des formes algébriques Trudy Tbilis. Mat. Inst. 9, 97–206 (1941) Janet, M.: Sur les systèmes d’équations aux dérivées partielles. J. Math. Pure Appl. 3, 65–151 (1920) Kandri-Rody, A., Kapur, D.: Computing the Gröbner basis of an ideal in polynomail rings over a Euclidean ring. J. Symb. Comput. 6, 37–56 (1990) Kandri-Rody, A., Weispfenning, W.: Non-commutative Gröbner Bases in Algebras of Solvable Type. J. Symb. Comput. 9, 1–26 (1990) Kredel, H.: Solvable Polynomial rings. Dissertation. Passau (1992) Levandovskyy, V.G.: Non-commutative Computer Algebra for Polynomial Algebras: Gröbner Bases, Applications and ImplementationDissertation, Kaiserslautern (2005) http://kluedo.ub.uni-kl.de/volltexte/2005/1883/ Levandovskyy, V.G.: PBW Bases, Non-Degeneracy Conditions and Applications In: Buchweitz, R.-O., Lenzing, H. (Eds.), Representation of Algebras and Related Topics (Proceedings of the ICRA X Conference), 45. AMS. Fields Institute Communications, pp. 229–246 Mora, F.: De Nugis Groebnerialium 4: Zacharias, Spears, Möller Proc. ISSAC’15. ACM, pp. 191–198 (2015) Mora, T.: Solving Polynomial Equation Systems 4 Vols., Cambridge University Press, I (2003), II (2005), III (2015), IV (2016) Mora, T.: Zacharias representation of effective associative rings. J. Symb. Comput. 99, 147–188 (2020) Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theor. Comput. Sci. 134(1), 131–173 (1994) Möller, H.M.: On the construction of Gröbner bases using syzygies. J. Symb. Comput. 6, 345–359 (1988) Nguefack, B., Pola, E.: Effective Buchberger–Zacharias–Weispfenning theory of skew polynomial extensions of restricted bilateral coherent rings. J. Symb. Comput. (in press) https://doi.org/10.1016/j.jsc.2019.03.003 Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933) Pan, L.: On the D-bases of polynomial ideals over principal ideal domains. J. Symb. Comput. 7, 55–69 (1988) Pesch, M.: Gröbner Bases in Skew Polynomial Rings. Dissertation, Passau (1997) Pesch, M.: Two-sided Gröbner bases in iterated ore extensions. Prog. Comput. Sci. Appl. Logic 15, 225–243 (1991) Pritchard, F.L.: A syzygies approach to non-commutative Gröbner bases. Preprint (1994) Pritchard, F.L.: The ideal membership problem in non-commutative polynomial rings. J. Symb. Comput. 22, 27–48 (1996) Reinert, B.: A systematic Study of Gröbner Basis Methods. Habilitation, Kaiserslautern (2003) Reinert, B.: Gröbner bases in function ring—a guide for introducing reduction relations to algebraic structures. J. Symb. Comput. 41, 1264–94 (2006) Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrass’schen Divisionsatz. Diplomarbait, Hamburg (1980) Schreyer, F.O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math. 421, 83–123 (1991) Schwartz, F.: Reduction and completion algorithm for partial differential equations. In: Proceedings of ISSAC’92. ACM, pp. 49–56 (1992) Seiler, W.M.: A combinatorial approach to involution and \(\delta \)-Regularity I: involutive bases in polynommial algebras of solvable type. J. AAECC 20, 207–259 (2009) Spear, D.A.: A constructive approach to commutative ring theory. In: Proceedings of the 1977 MACSYMA Users’ Conference, NASA CP-2012 , pp. 369–376 (1977) Tamari, D.: On a certain classification of rings and semigroups. Bull. Am. Math. Soc. 54, 153–158 (1948) Weispfenning, V.: Finite Gröbner bases in non-noetherian skew polynomial rings. In: Proceedings of ISSAC’92. ACM, pp. 320–332 (1992) Zacharias, G.: Generalized Gröbner bases in commutative polynomial rings, Bachelor’s thesis, MIT (1978) Zarkov, A.: Solving zero-dimensional involutive systems. Prog. Math. 143, 389–399 (1996) Zarkov, A., Blinkov, Y.: Involution approach to investing polynomial systems. Math. Comput. Simul. 42, 323–332 (1996)