Abstract
A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.
This material is based upon work supported by the National Science Foundation under Grant No. 1440534.
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References
Adrovic, D., Verschelde, J.: Computing Puiseux series for algebraic surfaces. In: van der Hoeven, J., van Hoeij, M. (eds.) Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC 2012), pp. 20–27. ACM (2012)
Adrovic, D., Verschelde, J.: Polyhedral methods for space curves exploiting symmetry applied to the cyclic n-roots problem. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 10–29. Springer, Heidelberg (2013)
Bernshteǐn, D.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)
Bogart, T., Jensen, A., Speyer, D., Sturmfels, B., Thomas, R.: Computing tropical varieties. J. Symbolic Comput. 42(1), 54–73 (2007)
Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Herrero, M., Jeronimo, G., Sabia, J.: Affine solution sets of sparse polynomial systems. J. Symbolic Comput. 51(1), 34–54 (2012)
Herrero, M., Jeronimo, G., Sabia, J.: Elimination for generic sparse polynomial systems. Discrete Comput. Geom. 51(3), 578–599 (2014)
Herrero, M., Jeronimo, G., Sabia, J.: Puiseux expansions and non-isolated points in algebraic varieties. Commun. Algebra 44(5), 2100–2109 (2016)
Hida, Y., Li, X., Bailey, D.: Algorithms for quad-double precision floating point arithmetic. In: 15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), 11–17, Vail, CO, USA, pp. 155–162. IEEE Computer Society (2001). Shortened version of Technical Report LBNL-46996, software at http://crd.lbl.gov/~dhbailey/mpdist/qd-2.3.9.tar.gz
Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64(212), 1541–1555 (1995). http://www.jstor.org/stable/2153370
Huber, B., Verschelde, J.: Polyhedral end games for polynomial continuation. Numer. Algorithms 18(1), 91–108 (1998)
Jensen, A., Leykin, A., Yu, J.: Computing tropical curves via homotopy continuation. Exp. Math. 25(1), 83–93 (2016)
Jensen, A., Yu, J.: Computing tropical resultants. J. Algebra 387, 287–319 (2013)
Jensen, A.: Computing Gröbner fans and tropical varieties in Gfan. In: Stillman, M., Takayama, N., Verschelde, J. (eds.) Software for Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol. 148, pp. 33–46. Springer, Heidelberg (2008)
Jeronimo, G., Matera, G., Solernó, P., Waissbein, A.: Deformation techniques for sparse systems. Found. Comput. Math. 9(1), 1–50 (2008). http://dx.doi.org/10.1007/s10208-008-9024-2
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)
OEIS Foundation Inc.: The on-line encyclopedia of integer sequences (2016). http://oeis.org. Accessed 03 Nov 2015
Römer, T., Schmitz, K.: Generic tropical varieties. J. Pure Appl. Algebra 216(1), 140–148 (2012). http://www.sciencedirect.com/science/article/pii/S0022404911001290
Sabeti, R.: Numerical-symbolic exact irreducible decomposition of cyclic-12. LMS J. Comput. Math. 14, 155–172 (2011)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)
Sommars, J., Verschelde, J.: Pruning algorithms for pretropisms of Newton polytopes. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 489–503 (2016)
Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999)
Verschelde, J.: Polyhedral methods in numerical algebraic geometry. In: Bates, D., Besana, G., Di Rocco, S., Wampler, C. (eds.) Interactions of Classical and Numerical Algebraic Geometry, Contemporary Mathematics, vol. 496, pp. 243–263. AMS (2009)
Verschelde, J.: Modernizing PHCpack through phcpy. In: de Buyl, P., Varoquaux, N. (eds.) Proceedings of the 6th European Conference on Python in Science (EuroSciPy 2013), pp. 71–76 (2014)
Verschelde, J., Verlinden, P., Cools, R.: Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal. 31(3), 915–930 (1994)
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Bliss, N., Verschelde, J. (2016). Computing All Space Curve Solutions of Polynomial Systems by Polyhedral Methods. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_6
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