Multiprojective witness sets and a trace test
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Jonathan D. Hauenstein
Abstract
In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalise the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.
Communicated by: M. Joswig
Funding: JDH was supported in part by NSF ACI 1460032 and Sloan Research Fellowship. JIR was supported in part by NSF DMS 1402545.
Acknowledgements
The authors want to thank Anton Leykin, Andrew Sommese, and Frank Sottile for helpful conversations, some of which occurred during the fall of 2014 at the Simons Institute for the Theory of Computing, leading to the results in this paper. The authors would also like to thank one anonymous referee for their detailed comments and suggestions which helped to significantly improve this paper. Certain aspects of the trace test first presented in earlier drafts of this paper have subsequently been reformulated in [26].
References
[1] H. Alt, Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkviereckes. Z. Angew. Math. Mech. 3 (1923), 13–19. JFM 49.0567.0310.1002/zamm.19230030103Suche in Google Scholar
[2] D. J. Bates, B. Davis, D. Eklund, E. Hanson, C. Peterson, Perturbed homotopies for finding all isolated solutions of polynomial systems. Appl. Math. Comput. 247 (2014), 301–311. MR3270842 Zbl 1338.1304610.1016/j.amc.2014.08.100Suche in Google Scholar
[3] D. J. Bates, J. D. Hauenstein, C. Peterson, A. J. Sommese, A numerical local dimension test for points on the solution set of a system of polynomial equations. SIAM J. Numer. Anal. 47 (2009), 3608–3623. MR2576513 Zbl 1211.1406610.1137/08073264XSuche in Google Scholar
[4] D. J. Bates, J. D. Hauenstein, A. J. Sommese, C. W. Wampler, Bertini: Software for numerical algebraic geometry. Available at bertini.nd.edu with permanent dx.doi.org/10.7274/R0H41PB5, 2006.Suche in Google Scholar
[5] D. J. Bates, J. D. Hauenstein, A. J. Sommese, C. W. Wampler, Numerically solving polynomial systems with Bertini, volume 25 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics, Philadelphia, 2013. MR3155500 Zbl 1295.6505710.1137/1.9781611972702Suche in Google Scholar
[6] D. A. Brake, J. D. Hauenstein, A. C. Liddell, Jr., Decomposing solution sets of polynomial systems using derivatives. In: Mathematical software–-ICMS 2016, volume 9725 of Lecture Notes in Comput. Sci., 127–135, Springer 2016. MR3662307 Zbl 0663065110.1007/978-3-319-42432-3_16Suche in Google Scholar
[7] D. A. Brake, J. D. Hauenstein, A. P. Murray, D. H. Myszka, C. W. Wampler, The complete solution of Alt–Burmester synthesis problems for four-bar linkages. J. Mechanisms Robotics8, issue 4, (2016).10.1115/1.4033251Suche in Google Scholar
[8] L. Chiantini, M. Mella, G. Ottaviani, One example of general unidentifiable tensors. J. Algebr. Stat. 5 (2014), 64–71. MR3279954 Zbl 1346.1412510.18409/jas.v5i1.25Suche in Google Scholar
[9] B. H. Dayton, Z. Zeng, Computing the multiplicity structure in solving polynomial systems. In: ISSAC’05, 116–123, ACM, New York 2005. MR2280537 Zbl 1360.6515110.1145/1073884.1073902Suche in Google Scholar
[10] E. Gawrilow, M. Joswig, polymake: a framework for analyzing convex polytopes. In: Polytopes–-combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Sem., 43–73, Birkhäuser, Basel 2000. MR1785292 Zbl 0960.6818210.1007/978-3-0348-8438-9_2Suche in Google Scholar
[11] D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at www.math.uiuc.edu/Macaulay2/.Suche in Google Scholar
[12] W. Hao, A. J. Sommese, Z. Zeng, Algorithm 931: an algorithm and software for computing multiplicity structures at zeros of nonlinear systems. ACM Trans. Math. Software40 (2013), Art. 5, 16. MR3118744 Zbl 1295.6505810.1145/2513109.2513114Suche in Google Scholar
[13] J. Hauenstein, J. I. Rodriguez, B. Sturmfels, Maximum likelihood for matrices with rank constraints. J. Algebr. Stat. 5 (2014), 18–38. MR3279952 Zbl 1345.6204310.18409/jas.v5i1.23Suche in Google Scholar
[14] J. D. Hauenstein, B. Mourrain, A. Szanto, Certifying isolated singular points and their multiplicity structure. In: ISSAC’15–-Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation, 213–220, ACM, New York 2015. MR3388302 Zbl 1345.6828610.1145/2755996.2756645Suche in Google Scholar
[15] J. D. Hauenstein, L. Oeding, G. Ottaviani, A. J. Sommese, Homotopy techniques for tensor decomposition and perfect identifiability. J. Reine Angew. Math. 753 (2019), 1–22. MR3987862 Zbl 0708968310.1515/crelle-2016-0067Suche in Google Scholar
[16] J. D. Hauenstein, A. J. Sommese, Witness sets of projections. Appl. Math. Comput. 217 (2010), 3349–3354. MR2733776 Zbl 1203.1407210.1016/j.amc.2010.08.067Suche in Google Scholar
[17] J. D. Hauenstein, A. J. Sommese, Membership tests for images of algebraic sets by linear projections. Appl. Math. Comput. 219 (2013), 6809–6818. MR3027848 Zbl 1285.1406610.1016/j.amc.2012.12.060Suche in Google Scholar
[18] J. D. Hauenstein, A. J. Sommese, C. W. Wampler, Regeneration homotopies for solving systems of polynomials. Math. Comp. 80 (2011), 345–377. MR2728983 Zbl 1221.6512110.1090/S0025-5718-2010-02399-3Suche in Google Scholar
[19] J. D. Hauenstein, A. J. Sommese, C. W. Wampler, Regenerative cascade homotopies for solving polynomial systems. Appl. Math. Comput. 218 (2011), 1240–1246. MR2831632 Zbl 1231.6519010.1016/j.amc.2011.06.004Suche in Google Scholar
[20] J. D. Hauenstein, C. W. Wampler, Isosingular sets and deflation. Found. Comput. Math. 13 (2013), 371–403. MR3047005 Zbl 1276.6502910.1007/s10208-013-9147-ySuche in Google Scholar
[21] J. D. Hauenstein, C. W. Wampler, Unification and extension of intersection algorithms in numerical algebraic geometry. Appl. Math. Comput. 293 (2017), 226–243. MR3549665 Zbl 1411.6507510.1016/j.amc.2016.08.023Suche in Google Scholar
[22] B. Huber, J. Verschelde, Polyhedral end games for polynomial continuation. Numer. Algorithms18 (1998), 91–108. MR1659862 Zbl 0933.6505710.1023/A:1019163811284Suche in Google Scholar
[23] J. Huh, B. Sturmfels, Likelihood geometry. In: Combinatorial algebraic geometry, volume 2108 of Lecture Notes in Math., 63–117, Springer 2014. MR3329087 Zbl 1328.1400410.1007/978-3-319-04870-3_3Suche in Google Scholar
[24] T. L. Lee, T. Y. Li, C. H. Tsai, HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing83 (2008), 109–133. MR2457354 Zbl 1167.6536610.1007/s00607-008-0015-6Suche in Google Scholar
[25] A. Leykin, Numerical algebraic geometry. J. Softw. Algebra Geom. 3 (2011), 5–10. MR2881262 Zbl 1311.1405710.2140/jsag.2011.3.5Suche in Google Scholar
[26] A. Leykin, J. I. Rodriguez, F. Sottile, Trace test. Arnold Math. J. 4 (2018), 113–125. MR3810571 Zbl 0697500610.1007/s40598-018-0084-3Suche in Google Scholar
[27] A. Leykin, J. Verschelde, A. Zhao, Newton’s method with deflation for isolated singularities of polynomial systems. Theoret. Comput. Sci. 359 (2006), 111–122. MR2251604 Zbl 1106.6504610.1016/j.tcs.2006.02.018Suche in Google Scholar
[28] E. Miller, B. Sturmfels, Combinatorial commutative algebra. Springer 2005. MR2110098 Zbl 1090.13001Suche in Google Scholar
[29] A. Morgan, A. Sommese, A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24 (1987), 101–113. MR914806 Zbl 0635.6505710.1016/0096-3003(87)90063-4Suche in Google Scholar
[30] A. P. Morgan, A. J. Sommese, Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29 (1989), 123–160. MR977815 Zbl 0664.6504910.1016/0096-3003(89)90099-4Suche in Google Scholar
[31] A. P. Morgan, A. J. Sommese, C. W. Wampler, A product-decomposition bound for Bezout numbers. SIAM J. Numer. Anal. 32 (1995), 1308–1325. MR1342295 Zbl 0854.6503810.1137/0732061Suche in Google Scholar
[32] J. R. Munkres, Topology. Prentice Hall, Upper Saddle River, NJ 2000. MR3728284 Zbl 0951.54001Suche in Google Scholar
[33] B. Shiffman, A. J. Sommese, Vanishing theorems on complex manifolds. Birkhäuser Boston, Boston MA 1985. MR782484 Zbl 0578.3205510.1007/978-1-4899-6680-3Suche in Google Scholar
[34] A. J. Sommese, J. Verschelde, C. W. Wampler, Numerical irreducible decomposition using projections from points on the components. In: Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), volume 286 of Contemp. Math., 37–51, Amer. Math. Soc. 2001. MR1874270 Zbl 1061.6859310.1090/conm/286/04753Suche in Google Scholar
[35] A. J. Sommese, J. Verschelde, C. W. Wampler, Using monodromy to decompose solution sets of polynomial systems into irreducible components. In: Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), volume 36 of NATO Sci. Ser. II Math. Phys. Chem., 297–315, Kluwer 2001. MR1866906 Zbl 0990.6505110.1007/978-94-010-1011-5_16Suche in Google Scholar
[36] A. J. Sommese, J. Verschelde, C. W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 40 (2002), 2026–2046 (2003). MR1974173 Zbl 1034.6503410.1137/S0036142901397101Suche in Google Scholar
[37] A. J. Sommese, C. W. Wampler, II, The numerical solution of systems of polynomials. World Scientific, Hackensack, NJ 2005. MR2160078 Zbl 1091.6504910.1142/5763Suche in Google Scholar
[38] J. Verschelde, Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25, 251–276 (1999).10.1145/317275.317286Suche in Google Scholar
[39] J. Verschelde, K. Gatermann, R. Cools, Mixed-volume computation by dynamic lifting applied to polynomial system solving. Discrete Comput. Geom. 16 (1996), 69–112. MR1397788 Zbl 0854.6811110.1007/BF02711134Suche in Google Scholar
[40] C. W. Wampler, A. P. Morgan, A. J. Sommese, Complete solution of the nine-point path synthesis problem for four-bar linkages. ASME J. Mech. Design114 (1992), 153–159.10.1115/1.2916909Suche in Google Scholar
[41] C. W. Wampler, A. J. Sommese, Numerical algebraic geometry and algebraic kinematics. Acta Numer. 20 (2011), 469–567. MR2805156 Zbl 1254.1303110.1017/S0962492911000067Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Multiprojective witness sets and a trace test
- Image Milnor number and 𝒜e-codimension for maps between weighted homogeneous irreducible curves
- Generalizations of 3-Sasakian manifolds and skew torsion
- Uniform modular lattices and affine buildings
- Homogeneous Finsler spaces with exponential metric
- Continuous CM-regularity of semihomogeneous vector bundles
- Solutions to the affine quasi-Einstein equation for homogeneous surfaces
- The geometry of H4 polytopes
Artikel in diesem Heft
- Frontmatter
- Multiprojective witness sets and a trace test
- Image Milnor number and 𝒜e-codimension for maps between weighted homogeneous irreducible curves
- Generalizations of 3-Sasakian manifolds and skew torsion
- Uniform modular lattices and affine buildings
- Homogeneous Finsler spaces with exponential metric
- Continuous CM-regularity of semihomogeneous vector bundles
- Solutions to the affine quasi-Einstein equation for homogeneous surfaces
- The geometry of H4 polytopes
