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Involutive divisions for effective involutive algorithms


Properties of involutive divisions on monomials are studied. A new method of involutive graphs is developed. The concept of complete global involutive division is introduced. A criterion of Noetherianity of involutive divisions, a property of graphs of global involutive division, a test for completeness of global involutive division, and a criterion of global involutive division are considered. A new series of involutive divisions is obtained by the process of completion. The properties of the divisions contained in the constructed series are studied. It is shown that the divisions from the series are better than the classical involutive divisions for involutive algorithms. A problem stated by Gao is solved: another series of involutive divisions is obtained. It is proved that all divisions of this series are continuous.

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  1. 1.

    A. V. Astrelin, O. D. Golubitsky, and E. V. Pankratiev, “Gröbner bases and involutive bases,” in: Algebra: Proceedings of the International Algebraic Conference on the Occasion of the 90th Birthday of A. G. Kurosh. Moscow, Russia, May 25–30, 1998, Walter de Gruyter, Berlin (2000), pp. 49–55.

    Google Scholar 

  2. 2.

    Chen Yu-Fu and Gao Xiao-Shan, “Involutive directions and new involutive divisions,” Comput. Math. Appl., 41, Nos. 7–8, 945–956 (2001).

    MathSciNet  Google Scholar 

  3. 3.

    V. P. Gerdt and Yu. A. Blinkov, “Involutive bases of polynomial ideals,” Math. Comput. Simulation, 45, 519–542 (1998).

    Article  MathSciNet  Google Scholar 

  4. 4.

    V. P. Gerdt and Yu. A. Blinkov, “Minimal involutive bases,” Math. Comput. Simulation, 45, 543–560 (1998).

    Article  MathSciNet  Google Scholar 

  5. 5.

    V. P. Gerdt, M. Berth, and G. Czichowski, Involutive divisions in “Mathematica”: Implementation and some applications.

  6. 6.

    M. Janet, “Sur les systèmes d’équations aux dérivées partielles,” J. Math. Pure Appl., 3, 65–151 (1920).

    MATH  Google Scholar 

  7. 7.

    J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York (1978).

    Google Scholar 

  8. 8.

    E. S. Shemyakova, “Involutive divisions. Graphs,” Programmirovanie, 30, No. 2 (2004).

  9. 9.

    J. Thomas, Differential Systems, American Mathematical Society, New York (1937).

    Google Scholar 

  10. 10.

    A. Yu. Zharkov and Yu. A. Blinkov, “Involutive systems of algebraic equation,” Programmirovanie, 53–56 (1994).

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 237–253, 2003.

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Shemyakova, E.S. Involutive divisions for effective involutive algorithms. J Math Sci 135, 3425–3436 (2006).

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