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Pruning Algorithms for Pretropisms of Newton Polytopes

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

Pretropisms are candidates for the leading exponents of Puiseux series that represent positive dimensional solution sets of polynomial systems. We propose a new algorithm to both horizontally and vertically prune the tree of edges of a tuple of Newton polytopes. We provide experimental results with our preliminary implementation in Sage that demonstrates that our algorithm compares favorably to the definitional algorithm.

This material is based upon work supported by the National Science Foundation under Grant No. 1440534.

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References

  1. 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)

    Google Scholar 

  2. 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)

    Chapter  Google Scholar 

  3. Assarf, B., Gawrilow, E., Herr, K., Joswig, M., Lorenz, B., Paffenholz, A., Rehn, T.: Computing convex hulls and counting integer points with polymake. arXiv:1408.4653v2

  4. Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8(3), 295–313 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Backelin, J.: Square multiples n give infinitely many cyclic n-roots. Reports, Matematiska Institutionen 8, Stockholms universitet (1989)

    Google Scholar 

  6. Bagnara, R., Hill, P., Zaffanella, E.: The Parma Polyhedral Library: toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program. 72(1–2), 3–21 (2008)

    Article  MathSciNet  Google Scholar 

  7. Bernshteǐn, D.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)

    Article  MathSciNet  Google Scholar 

  8. Bjöck, G., Saffari, B.: New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries. C.R. Acad. Sci. Paris Série I 320, 319–324 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Bliss, N., Verschelde, J.: Computing all space curve solutions of polynomial systems by polyhedral methods. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 73–86. Springer, Heidelberg (2016)

    Google Scholar 

  10. Bogart, T., Jensen, A., Speyer, D., Sturmfels, B., Thomas, R.: Computing tropical varieties. J. Symbolic Comput. 42(1), 54–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Braun, V., Hampton, M.: Polyhedra module of Sage, The Sage Development Team (2011)

    Google Scholar 

  12. Büeler, B., Enge, A., Fukuda, K.: Exact volume computation for polytopes:a practical study. In: Kalai, G., Ziegler, G. (eds.) Polytopes -Combinatorics and Computation, DMV Seminar, vol. 29, pp. 131–154. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  13. De Loera, J., Rambau, J., Santos, F.: Triangulations, Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25. Springer, Heidelberg (2010)

    MATH  Google Scholar 

  14. Emiris, I.: Sparse Elimination and Applications in Kinematics. Ph.D. thesis, University of California at Berkeley, Berkeley (1994)

    Google Scholar 

  15. Emiris, I., Canny, J.: Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symbolic Comput. 20(2), 117–149 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Emiris, I., Fisikopoulos, V.: Efficient random-walk methods for approximating polytope volume. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry (SoCG 2014), pp. 318–327. ACM (2014)

    Google Scholar 

  17. Emiris, I., Fisikopoulos, V., Gärtner, B.: Efficient edge-skeleton computation for polytopes defined by oracles. J. Symbolic Comput. 73, 139–152 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Emiris, I., Fisikopoulos, V., Konaxis, C.: Exact and approximate algorithms for resultant polytopes. In: Proceedings of the 28th European Workshop on Computational Geometry (EuroCG 2012) (2012)

    Google Scholar 

  19. Führ, H., Rzeszotnik, Z.: On biunimodular vectors for unitary matrices. Linear Algebra Appl. 484, 86–129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fukuda, K., Prodon, A.: Double description method revisited. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 91–111. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  21. Gao, T., Li, T.: Mixed volume computation for semi-mixed systems. Discrete Comput. Geom. 29(2), 257–277 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gao, T., Li, T., Wu, M.: Algorithm 846: MixedVol: a software package for mixed-volume computation. ACM Trans. Math. Softw. 31(4), 555–560 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hampton, M., Jensen, A.: Finiteness of relative equilibria in the planar generalized n-body problem with fixed subconfigurations. J. Geom. Mech. 7(1), 35–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hampton, M., Moeckel, R.: Finiteness of stationary configurations of the four-vortex problem. Trans. Am. Math. Soci. 361(3), 1317–1332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jensen, A.: Gfan, a software system for Gröbner fans and tropical varieties. http://home.imf.au.dk/jensen/software/gfan/gfan.html

  26. 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)

    Chapter  Google Scholar 

  27. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  28. Malajovich, G.: Computing mixed volume and all mixed cells in quermassintegral time, to appear in Found. Comput. Math. http://dx.doi.org/10.1007/s10208-016-9320-1

  29. Maurer, J.: Puiseux expansion for space curves. Manuscripta Math. 32, 91–100 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mizutani, T., Takeda, A.: DEMiCs: a software package for computing the mixed volume via dynamic enumeration of all mixed cells. In: Stillman, M., Takayama, N., Verschelde, J. (eds.) Software for Algebraic Geometry. The IMA Volumes in Mathematics and Its Applications, vol. 148, pp. 59–79. Springer, New York (2008)

    Chapter  Google Scholar 

  31. Mizutani, T., Takeda, A., Kojima, M.: Dynamic enumeration of all mixed cells. Discrete Comput. Geom. 37(3), 351–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Novoseltsev, A.: lattice_polytope module of Sage, The Sage Development Team (2011)

    Google Scholar 

  33. R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2008). http://www.R-project.org, ISBN 3-900051-07-0

  34. Sommars, J., Verschelde, J.: Exact gift wrapping to prune the tree of edges of Newton polytopes to compute pretropisms. arXiv:1512.01594

  35. Sommars, J., Verschelde, J.: Computing pretropisms for the cyclic n-roots problem. In: 32nd European Workshop on Computational Geometry (EuroCG 2016), pp. 235–238 (2016)

    Google Scholar 

  36. Stein, W., et al.: Sage Mathematics Software (Version 6.9). The Sage Development Team (2015). http://www.sagemath.org

  37. Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999)

    Article  MATH  Google Scholar 

  38. 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)

    Google Scholar 

  39. Ziegler, G.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street (m/c 249), Chicago, IL, 60607-7045, USA

    Jeff Sommars & Jan Verschelde

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  1. Jeff Sommars
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  2. Jan Verschelde
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Correspondence to Jeff Sommars .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Sommars, J., Verschelde, J. (2016). Pruning Algorithms for Pretropisms of Newton Polytopes. 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_31

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_31

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