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Computing the Dixon resultant with the Maple package DR. (English) Zbl 1388.68319
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 273-287 (2017).
Summary: The Maple package DR provides functions for computing the Dixon resultant of a system of parametric multi-variate polynomials. The Dixon resultant constitutes a necessary condition for the polynomials to have a common root after specializing their parameters. The newest version 2 of the package DR includes the new heuristic pivot row detection of factors for extracting the Dixon resultant from the Dixon matrix. It is shown to be efficient on systems of benchmark polynomials, outperforming other heuristics for a majority of systems.
For the entire collection see [Zbl 1379.13001].

MSC:
68W30 Symbolic computation and algebraic computation
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
15A15 Determinants, permanents, traces, other special matrix functions
Software:
DR; Fermat; Maple; SageMath; SDeval
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References:
[1] 1. Bus, L., Elkadi, M., Mourrain, B.: Using projection operators in computer aided geometric design. Comtempor. Math. 334 , 321-342 (2003) · Zbl 1036.14500
[2] 2. Chionh, E.-W., Zhang, M., Goldman, R.N.: Fast computation of the Bezout and Dixon resultant matrices. J. Symb. Comput. 33 (1), 13-29 (2002) · Zbl 0996.65046
[3] 3. Chtcherba, A.D., Kapur, D.: Conditions for determinantal formula for resultant of a polynomial system. In: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, pp. 55-62. ACM (2006) · Zbl 1356.13043
[4] 4. Chtcherba, A., Kapur, D., Minimair, M.: Cayley-Dixon projection operator for multi-univariate composed polynomials. J. Symb. Comput. 44 (8), 972-999 (2009) · Zbl 1234.68470
[5] 5. Coutsias, E.A., Seok, C., Jacobson, M.P., Dill, K.A.: A kinematic view of loop closure. J. Comput. Chem. 25 , 510-528 (2004)
[6] 6. Dixon, A.L.: The eliminant of three quantics in two independent variables. Proc. Lond. Math. Soc. 7 (49-69), 473-492 (1908) · JFM 40.0207.01
[7] 7. Emiris, I.Z., Mourrain, B.: Matrices in elimination theory. J. Symb. Comput. 28 (12), 3-44 (1999) · Zbl 0943.13005
[8] 8. Heinle, A., Levandovskyy, V.: The SDEval benchmarking toolkit. ACM Commun. Comput. Algebr. 49 (1), 1-9 (2015) · Zbl 1365.68493
[9] 9. Hu, H.Y., Wang, Z.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, New York (2002) · Zbl 1035.93002
[10] 10. Kapur, D., Minimair, M.: Multivariate resultants in Bernstein basis. In: Proceedings of the 7th International Conference on Automated Deduction in Geometry. Lecture Notes in Computer Science, vol. 6301, pp. 60-85. Springer, Shanghai (2011) · Zbl 1302.13029
[11] 11. Kapur, D., Saxena, T. Yang, L.: Algebraic and geometric reasoning using dixon resultants. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC ’94), pp. 99-107. ACM, New York (1994) · Zbl 0964.68536
[12] 12. Lewis, R.H.: Comparing acceleration techniques for the Dixon and Macaulay resultants. Math. Comput. Simul. (2008) · Zbl 1192.65056
[13] 13. Lewis R.H.: Heuristics to accelerate the Dixon resultant. Math. Comput. Simul. 77 (4), 400-407 (2008) · Zbl 1138.65037
[14] 14. Lewis, R.H.: Comparison of GCD in several systems.
[15] 15. Lewis, R.H.: Dixon resultant computation in the Fermat system.
[16] 16. Lewis, R.H.: Parametric polynomial system motivated by Bricard.
[17] 17. Lewis, R.H., Stiller, P.: Solving the recognition problem for six lines using the Dixon resultant. Math. Comput. Simul. 49 (3), 205-219 (1999)
[18] 18. Little, J.B.: Solving the SelesnickBurrus filter design equations using computational algebra and algebraic geometry. Adv. Appl. Math. 31 (2), 463-500 (2003) · Zbl 1032.94002
[19] 19. Minimair, M.: DR: Maple package for Dixon resultant computation (2015). · Zbl 1388.68319
[20] 20. Nakos, G., Williams, R.M.: Elimination with the Dixon resultant. Math. Educ. Res. 6 (3), 11-21 (1997)
[21] 21. Paláncz, B.: Application of Dixon resultant to satellite trajectory control by pole placement. J. Symb. Comput. 50 , 79-99 (2013) · Zbl 1253.93051
[22] 22. Paláncz, B., Zaletnyik, P., Awange, J.L., Grafarend, E.W.: Dixon resultants solution of systems of geodetic polynomial equations. J. Geodesy 82 (8), 505-511 (2007)
[23] 23. Stein, W.: Sage—open-source mathematical software system (2008)
[24] 24. Sun, W.K.: Solving 3-6 parallel robots by Dixon resultant. Appl. Mech. Mater. 235 , 158-163 (2012)
[25] 25. Zhao, S., Fu, H.: An extended fast algorithm for constructing the Dixon resultant matrix. Sci. China Ser. A Math. 48 (1), 131-143 (2005) · Zbl 1122.14302
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