Abstract
In the present paper we consider a class of involutive monomial divisions pairwise constructed by the partition of variables into multiplicative and nonmultiplicative generated by a total monomial ordering. If this ordering is admissible or the inverse of an admissible ordering, then the involutive division generated possesses all algorithmically important properties such as continuity, constructivity, and noetherianity. Among all such divisions, we single out those generated by antigraded monomial orderings. We demonstrate, by example of the antigraded lexicographic ordering, that the divisions of this class are heuristically better than the classical Janet division. The last division is pairwise generated by the pure lexicographic ordering and up to now has been considered as computationally best.
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References
Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Mathematics and Computers in Simulation 45, 519–542 (1998); Minimal involutive bases, ibid, 543–560
Apel, J.: The theory of involutive divisions and an application to Hilbert function computations. J. Symbolic Computation 25, 683–704 (1998)
Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles. Cahiers Scientifiques, IV, Gauthier-Villars, Paris (1929)
Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Computational Commutative and Non-Commutative Algebraic Geometry, pp. 199–225. IOS Press, Amsterdam (2005)
Seiler, W.M.: Involution: The formal theory of differential equations and its applications in computer algebra. In: Algorithms and Computation in Mathematics, vol. 24. Springer, Heidelberg (2010)
Gerdt, V.P., Blinkov, Y. A.: Specialized computer algebra system GINV. Programming and Computer Software 34(2), 112–123 (2008)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-2 - A computer algebra system for polynomial computations (2010), http://www.singular.uni-kl.de
Gerdt, V.P.: Involutive division technique: some generalizations and optimizations. J. Math. Sciences 108(6), 1034–1051 (2002)
Chen, Y.-F., Gao, X.-S.: Involutive directions and new involutive divisions. Computers and Mathematics with Applications 41, 945–956 (2001)
Semenov, A.S.: On connection between constructive involutive divisions and monomial orderings. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 261–278. Springer, Heidelberg (2006)
Semenov, A.S.: Constructivity of involutive divisions. Programming and Computer Software 32(2), 96–102 (2007)
Semenov, A.S., Zyuzikov, P.A.: Involutive divisions and monomial orderings. Programming and Computer Software 33(3), 139–146 (2007); Involutive divisions and monomial orderings: Part II. Ibid 34(2), 107–111 (2008)
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Graduate Texts in Mathematics, vol. 185. Springer, New York (2005)
Greul, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2007)
Becker, T., Weispfenning, V.: Gröbner Bases. A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)
Gerdt, V.P.: On the relation between Pommaret and Janet bases. In: Computer Algebra in Scientific Computing / CASC 2000, pp. 167–181. Springer, Berlin (2000)
Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Thomas decomposition of algebraic and differential systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 31–54. Springer, Heidelberg (2010)
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Gerdt, V.P., Blinkov, Y.A. (2011). Involutive Division Generated by an Antigraded Monomial Ordering. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_13
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DOI: https://doi.org/10.1007/978-3-642-23568-9_13
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