Abstract
We recall definitions and properties of parametric solvable polynomial rings and variants. For recursive solvable polynomial rings, i.e. solvable polynomial rings with coefficients from a solvable polynomial ring, also commutator relations between main and coefficient variables are introduced. From these rings solvable quotient and residue class rings can be constructed and used as coefficients of solvable polynomials. The resulting skew extension fields of the ground field can be applied for skew root finding or primary ideal decomposition. We present the design and implementation of these rings in the strongly typed, generic, object oriented computer algebra system JAS.
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References
Kredel, H.: On a Java Computer Algebra System, its performance and applications. Science of Computer Programming 70(2–3), 185–207 (2008)
Kredel, H.: The Java algebra system (JAS). Technical report (since 2000). http://krum.rz.uni-mannheim.de/jas/
Kredel, H.: Unique factorization domains in the Java computer algebra system. In: Sturm, T., Zengler, C. (eds.) ADG 2008. LNCS, vol. 6301, pp. 86–115. Springer, Heidelberg (2011)
Kredel, H., Jolly, R.: Algebraic structures as typed objects. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 294–308. Springer, Heidelberg (2011)
Kredel, H.: Distributed Gröbner bases computation with MPJ. In: IEEE AINA Workshops, Barcelona, Spain, pp. 1429–1435 (2013)
Apel, J., Lassner, W.: An algorithm for calculations in eveloping fields of Lie algebras. In: Proc. Int. Conf. on Computer Algebra and its Applications in Theoretical Physics, JINR, D11–85-791, Dubna (1985)
Apel, J., Lassner, W.: Computation and simplification in Lie fields. In: EUROCAL 1987, pp. 468–478 (1987)
Kandri Rody, A., Weispfennning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comp. 9(1), 1–26 (1990)
Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theor. Comput. Sci. 134(1), 131–173 (1994)
Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14(1), 1–29 (1992)
Kredel, H., Weispfenning, V.: Parametric Gröbner bases for non-commutative polynomials. In: Shirkov, D.V., Rostovtsev, V.A., Gerdt, V.P. (eds.) IV. Int. Conf. on Computer Algebra in Physical Research, Dubna, USSR, 1990, pp. 236–246. World Scientific Publishing (1991)
Kredel, H.: Solvable Polynomial Rings. Dissertation, Universität Passau, Passau (1992)
Bueso, J.L., Gómez-Torrecillas, J., Verschoren, A.: Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups. Kluwer Academic Publishers (2003)
Levandovskyy, V., Schönemann, H.: Plural: a computer algebra system for noncommutative polynomial algebras. In: Proc. Symbolic and Algebraic Computation, International Symposium ISSAC 2003, Philadelphia, USA, pp. 176–183 (2003)
Levandovskyy, V.: Non-commutative Computer Algebra for polynomial algebras: Gröbner bases, applications and implementation. Dissertation, University of Kaiserslautern, Kaiserslautern (2005)
Cohn, P.M.: Skew Fields, Theory of General Division Rings. Encyclopedia of Mathematics and its Applications, vol. 57. Cambridge University Press (1995)
Kredel, H.: On the design of a Java computer algebra system. In: Proc. PPPJ 2006, University of Mannheim, pp. 143–152 (2006)
Kredel, H., Pesch, M.: MAS: the modula-2 algebra system. In: Computer Algebra Handbook, pp. 421–428. Springer (2003)
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bull. 31(2), 2–9 (1997)
Montes, A.: An new algorithm for discussing Gröbner basis with parameters. J. Symb. Comput. 33(1–2), 183–208 (2002)
Suzuki, A., Sato, Y.: An alternative approach to comprehensive Gröbner bases. J. Symb. Comput. 36(3–4), 649–667 (2003)
Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Proc. ISSAC 2007, pp. 299–306 (2007)
Inoue, S., Sato, Y.: On the parallel computation of comprehensive Gröbner systems. In: Proc. PASCO 2007, pp. 99–101 (2007)
Kredel, H.: Comprehensive Gröbner bases in a Java computer algebra system. In: Proceedings ASCM 2009, Kyushu University, Fukuoka, Japan, pp. 77–90 (2009)
Kredel, H.: Comprehensive Gröbner bases in a Java computer algebra system. In: Feng, R., Lee, W.S., Sato, Y. (eds.) Computer Mathematics, pp. 93–108. Springer, Heidelberg (2014)
Nabeshima, K.: PGB: a package for computing parametric polynomial systems. In: Proceedings ASCM 2009, Kyushu University, Fukuoka, Japan, pp. 111–122 (2009)
Greuel, G., Pfister, G., Schönemann, H.: Singular - a computer algebra system for polynomial computations. In: Computer Algebra Handbook, pp. 445–450. Springer (2003)
Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symb. Comput. 43(8), 515–544 (2008)
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Kredel, H. (2015). Parametric Solvable Polynomial Rings and Applications. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_21
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DOI: https://doi.org/10.1007/978-3-319-24021-3_21
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