Nearest points on toric varieties

  • Martin Helmer
  • Bernd Sturmfels

Abstract

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.

References

Supplementary website, https://math.berkeley.edu/~mhelmer/Software/toricED, copy at http://martin-helmer.com/Software/toricED.html, code also at https://arxiv.org/src/1603.06544v4/anc.

Aluffi, P., Projective duality and a Chern-Mather involution, Trans. Amer. Math. Soc. 370 (2018), no. 3, 1803-1822. https://doi.org/10.1090/tran/7042

Bank, B., Giusti, M., Heintz, J., and Mbakop, G. M., Polar varieties and efficient real elimination, Math. Z. 238 (2001), no. 1, 115–144. https://doi.org/10.1007/PL00004896

Bank, B., Giusti, M., Heintz, J., and Pardo, L. M., Generalized polar varieties: geometry and algorithms, J. Complexity 21 (2005), no. 4, 377–412. https://doi.org/10.1016/j.jco.2004.10.001

Bates, D. J., Hauenstein, J. D., Sommese, A. J., and Wampler, C. W., Numerically solving polynomial systems with Bertini, Software, Environments, and Tools, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.

Brasselet, J.-P., Seade, J., and Suwa, T., Vector fields on singular varieties, Lecture Notes in Mathematics, vol. 1987, Springer-Verlag, Berlin, 2009. https://doi.org/10.1007/978-3-642-05205-7

Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. https://doi.org/10.1090/gsm/124

Dickenstein, A., Feichtner, E. M., and Sturmfels, B., Tropical discriminants, J. Amer. Math. Soc. 20 (2007), no. 4, 1111–1133. https://doi.org/10.1090/S0894-0347-07-00562-0

Dickenstein, A. and Tabera, L. F., Singular tropical hypersurfaces, Discrete Comput. Geom. 47 (2012), no. 2, 430–453. https://doi.org/10.1007/s00454-011-9364-6

Draisma, J., Horobeţ, E., Ottaviani, G., Sturmfels, B., and Thomas, R. R., The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16 (2016), no. 1, 99–149. https://doi.org/10.1007/s10208-014-9240-x

Ernström, L., A Plücker formula for singular projective varieties, Comm. Algebra 25 (1997), no. 9, 2897–2901. https://doi.org/10.1080/00927879708826029

Esterov, A., Newton polyhedra of discriminants of projections, Discrete Comput. Geom. 44 (2010), no. 1, 96–148. https://doi.org/10.1007/s00454-010-9242-7

Fulton, W., Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-1-4612-1700-8

Gel'fand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. https://doi.org/10.1007/978-0-8176-4771-1

Grayson, D. and Stillman, M., Macaulay2, a software system for research in algebraic geometry, availbale at www.math.uiuc.edu/Macaulay2/.

Holme, A., The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Math. 61 (1988), no. 2, 145–162. https://doi.org/10.1007/BF01259325

Horobeţ, E., The data singular and the data isotropic loci for affine cones, Comm. Algebra 45 (2017), no. 3, 1177–1186. https://doi.org/10.1080/00927872.2016.1172632

Huggins, P., Sturmfels, B., Yu, J., and Yuster, D. S., The hyperdeterminant and triangulations of the $4$-cube, Math. Comp. 77 (2008), no. 263, 1653–1679. https://doi.org/10.1090/S0025-5718-08-02073-5

Kleiman, S. L., Tangency and duality, in “Proceedings of the 1984 Vancouver conference in algebraic geometry”, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 163-225.

Maclagan, D. and Sturmfels, B., Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015.

MacPherson, R. D., Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. https://doi.org/10.2307/1971080

Matsui, Y. and Takeuchi, K., A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties, Adv. Math. 226 (2011), no. 2, 2040–2064. https://doi.org/10.1016/j.aim.2010.08.020

Nødland, B., Singular toric varieties, Master's thesis, University of Oslo, Norway, 2015.

Ottaviani, G., Spaenlehauer, P.-J., and Sturmfels, B., Exact solutions in structured low-rank approximation, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1521–1542. https://doi.org/10.1137/13094520X

Piene, R., Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 247–276.

Piene, R., Cycles polaires et classes de Chern pour les variétés projectives singulières, in “Introduction à la théorie des singularités, II”, Travaux en Cours, vol. 37, Hermann, Paris, 1988, pp. 7-34.

Piene, R., Chern-Mather classes of toric varieties, preprint arXiv:16030285v1, 2016.

Safey El Din, M. and Schost, É., Polar varieties and computation of one point in each connected component of a smooth algebraic set, in “Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation”, ACM, New York, 2003, pp. 224--231. https://doi.org/10.1145/860854.860901

Stanley, R. P., Eulerian partitions of a unit hypercube, in “Higher combinatorics, Proceedings of the NATO Advanced Study Institute held in Berlin, September 1–10, 1976” (Aigner, M., ed.), NATO Advanced Study Institute Series. Ser. C: Mathematical and Physical Sciences, vol. 31, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1977, p. 49.

Sturmfels, B., Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996.

Ziegler, G. M., Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4613-8431-1

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Published
2018-04-08
How to Cite
Helmer, M., & Sturmfels, B. (2018). Nearest points on toric varieties. MATHEMATICA SCANDINAVICA, 122(2), 213-238. https://doi.org/10.7146/math.scand.a-101478
Section
Articles