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Lopsided approximation of amoebas. (English) Zbl 1410.13019

The log absolute map is the map \((\mathbb{C}\ast)^n\to\mathbb{R}^n\) defined as \((z_1,\dots,z_n)\mapsto(\log(|z_1|,\dots,\log(|z_n|)\). The image of a hypersurface defined by a Laurent polynomial \(f\) is called the amoeba of \(f\). The paper [K. Purbhoo, Duke J. Math. 121, No. 3, 407–455 (2008; Zbl 1233.14036)] gives an algorithm to decide if a given point in \(\mathbb{R}^n\) is contained in the amoeba of a given Laurent polynomial \(f\). An essential step is the computation of the cyclic resultant of \(f\). This paper gives an efficient algorithm for computing the cyclic resultant.

MSC:

13P15 Solving polynomial systems; resultants
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
14T05 Tropical geometry (MSC2010)
90C59 Approximation methods and heuristics in mathematical programming
90C90 Applications of mathematical programming

Citations:

Zbl 1233.14036

Software:

SageMath; SINGULAR
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Avenda\~no, Mart\'\i n.; Kogan, Roman; Nisse, Mounir; Rojas, J. Maurice, Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces, J. Complexity, 46, 45-65 (2018) · Zbl 1420.14135 · doi:10.1016/j.jco.2017.11.008
[2] Bogdanov, D. V.; Kytmanov, A. A.; Sadykov, T. M., Algorithmic computation of polynomial amoebas. Computer algebra in scientific computing, Lecture Notes in Comput. Sci. 9890, 87-100 (2016), Springer, Cham · Zbl 1453.14145 · doi:10.1007/978-3-319-45641-6\_7
[3] Basu, Saugata; Pollack, Richard; Roy, Marie-Fran\c{c}oise, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics 10, x+662 pp. (2006), Springer-Verlag, Berlin · Zbl 1102.14041
[4] Cooley, James W.; Tukey, John W., An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19, 297-301 (1965) · Zbl 0127.09002 · doi:10.2307/2003354
[5] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 6.4.1), 2016, http://www.sagemath.org.
[6] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch\"onemann, Singular 4-0-2 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2015.
[7] T. de Wolff, Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits, Oberwolfach Rep., 23, 1308-1311 (2015), European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach
[8] Forsberg, Mikael; Passare, Mikael; Tsikh, August, Laurent determinants and arrangements of hyperplane amoebas, Adv. Math., 151, 1, 45-70 (2000) · Zbl 1002.32018 · doi:10.1006/aima.1999.1856
[9] Gel\cprime fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, x+523 pp. (1994), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0827.14036 · doi:10.1007/978-0-8176-4771-1
[10] Hillar, Christopher J., Cyclic resultants, J. Symbolic Comput., 39, 6, 653-669 (2005) · Zbl 1130.12002 · doi:10.1016/j.jsc.2005.01.001
[11] Hillar, Christopher J.; Levine, Lionel, Polynomial recurrences and cyclic resultants, Proc. Amer. Math. Soc., 135, 6, 1607-1618 (2007) · Zbl 1131.11011 · doi:10.1090/S0002-9939-06-08672-2
[12] Iliman, Sadik; de Wolff, Timo, Amoebas, nonnegative polynomials and sums of squares supported on circuits, Res. Math. Sci., 3, Paper No. 9, 35 pp. (2016) · Zbl 1415.11071 · doi:10.1186/s40687-016-0052-2
[13] Mikhalkin, G., Real algebraic curves, the moment map and amoebas, Ann. of Math. (2), 151, 1, 309-326 (2000) · Zbl 1073.14555 · doi:10.2307/121119
[14] Mikhalkin, Grigory, Amoebas of algebraic varieties and tropical geometry. Different faces of geometry, Int. Math. Ser. (N. Y.) 3, 257-300 (2004), Kluwer/Plenum, New York · Zbl 1072.14013 · doi:10.1007/0-306-48658-X\_6
[15] A. E. Pellet, Sur un mode de s\'eparation des racines des \'equations et la formule de Lagrange, Darboux Bull. (2) 5 (1881), 393-395, In French. · JFM 13.0074.01
[16] Passare, Mikael; Pochekutov, Dmitry; Tsikh, August, Amoebas of complex hypersurfaces in statistical thermodynamics, Math. Phys. Anal. Geom., 16, 1, 89-108 (2013) · Zbl 1281.82010 · doi:10.1007/s11040-012-9122-x
[17] Passare, Mikael; Rullg\aa rd, Hans, Amoebas, Monge-Amp\`ere measures, and triangulations of the Newton polytope, Duke Math. J., 121, 3, 481-507 (2004) · Zbl 1043.32001 · doi:10.1215/S0012-7094-04-12134-7
[18] Passare, Mikael; Tsikh, August, Amoebas: their spines and their contours. Idempotent mathematics and mathematical physics, Contemp. Math. 377, 275-288 (2005), Amer. Math. Soc., Providence, RI · Zbl 1079.32008 · doi:10.1090/conm/377/06997
[19] Purbhoo, Kevin, A Nullstellensatz for amoebas, Duke Math. J., 141, 3, 407-445 (2008) · Zbl 1233.14036 · doi:10.1215/00127094-2007-001
[20] H. Rullgrd, Topics in geometry, analysis and inverse problems, Ph.D. thesis, Stockholm University, 2003.
[21] F. Schroeter and T. de Wolff, The boundary of amoebas, 2013, Preprint, arXiv:1310.7363.
[22] Theobald, Thorsten; de Wolff, Timo, Approximating amoebas and coamoebas by sums of squares, Math. Comp., 84, 291, 455-473 (2015) · Zbl 1318.14054 · doi:10.1090/S0025-5718-2014-02828-7
[23] Theobald, Thorsten, Computing amoebas, Experiment. Math., 11, 4, 513-526 (2003) (2002) · Zbl 1100.14048
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