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Comparing acceleration techniques for the Dixon and Macaulay resultants. (English) Zbl 1192.65056

Summary: The Bézout-Dixon resultant method [cf. A. L. Dixon, Lond. M. S. Proc. (2) {6}, 468–478 (1908; Zbl JFM 39.0219.02)] for solving systems of polynomial equations lends itself to various heuristic acceleration techniques, previously reported by the present author, which can be extraordinarily effective. In this paper, we discuss how well these techniques apply to the Macaulay resultant [cf. D. Cox, J. Little and D. O’Shea, Using algebraic geometry. Graduate Texts in Mathematics. 185. New York, NY: Springer (1998; Zbl 0920.13026)]. In brief, we find that they do work there with some difficulties, but the Dixon method is greatly superior. That they work at all is surprising and begs theoretical explanation.

MSC:

65H10 Numerical computation of solutions to systems of equations
65H04 Numerical computation of roots of polynomial equations
13P15 Solving polynomial systems; resultants

Software:

MR; Fermat
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References:

[1] P. Bikker, On Bezout’s method for computing the resultant. RISC-Linz Report Series, Johannes Kepler University, Linz, Austria, 1995.
[2] Buse, L.; Elkadi, M.; Mourrain, B., Generalized resultants over unirational algebraic varieties, Journal of symbolic computations, 29, 515-526, (2000) · Zbl 0983.14039
[3] Cox, D.; Little, J.; O’Shea, D., Using algebraic geometry. graduate texts in mathematics, 185, (1998), Springer-Verlag New York
[4] Dixon, A.L., The eliminant of three quantics in two independent variables, Proceedings of the London mathematics society, 6, 468-478, (1908) · JFM 39.0219.02
[5] Emiris, I.Z.; Canny, J.F., Efficient incremental algorithms for the sparse resultant and the mixed volume, Journal of symbolic computations, 20, 2, 117-149, (1995) · Zbl 0843.68036
[6] Havel, T.F., Some examples of the use of distances as coordinates for Euclidean geometry, Journal of symbolic computations, 11, 579-593, (1991) · Zbl 0743.51011
[7] Kapur, D.; Saxena, T.; Yang, L., ()
[8] Kotsireas, I.S.; Karamanos, K., Exact computation of the bifurcation point B4 of the logistic map and the bailey – broadhurst conjectures, International journal of bifurcation and chaos, 14, 7, 2417-2423, (2004) · Zbl 1077.37511
[9] Lewis, R.H., Heuristics to accelerate the Dixon resultant, Mathematics and computers in simulation, 77, 4, 400-407, (2008) · Zbl 1138.65037
[10] R.H. Lewis, Computer Algebra System Fermat. http://home.bway.net/lewis/ (2008).
[11] R.H. Lewis, Using the Dixon resultant on big problems, in: CBMS, Conference, Texas, A.M., University, May 2002. http://www.math.tamu.edu/conferences/cbms/abs.html
[12] Lewis, R.H., Exploiting symmetry in a polynomial system with the Dixon resultant, international conference on applications of computer algebra (ACA), (2004), Lamar University Texas, July
[13] Lewis, R.H.; Bridgett, S., Conic tangency equations arising from apollonius problems in biochemistry, Mathematics and computers in simulation, 61, 2, 101-114, (2003) · Zbl 1016.51014
[14] Lewis, R.H.; Coutsias, E.A., Algorithmic search for flexibility using resultants of polynomial systems, () · Zbl 1195.68115
[15] M. Minimair, Macaulay Resultant Package. http://minimair.org/software/ (2007).
[16] D. Robertz, V. Gerdt, comparison of software systems, http://home.bway.net/lewis/fermat/gcdcomp (2005).
[17] Sommese, A.; Verschelde, J.; Wampler, C., Numerical decomposition of the solution sets of polynomial systems into irreducible components, Sinum, 38, 6, 2022-2046, (2001) · Zbl 1002.65060
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