nach oben

Programming and Computer Software

Erschienen in:

01.04.2023

Computing the Connected Components of the Complement to the Amoeba of a Polynomial in Several Complex Variables

verfasst von: T. A. Zhukov, T. M. Sadykov

Erschienen in: Programming and Computer Software | Ausgabe 2/2023

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we propose a method for computing and visualizing the amoeba of a Laurent polynomial in several complex variables, which is applicable in arbitrary dimension. The algorithms developed based on this method are implemented as a free web service (http://amoebas.ru), which enables interactive computation of amoebas for polynomials in two variables, as well as provides a set of precomputed amoebas and their cross-sections in higher dimensions. The correctness and running time of the proposed algorithms are tested against a set of optimal polynomials in two, three, and four variables, which are generated using Mathematica computer algebra system. The developed program code makes it possible, in particular, to generate optimal hypergeometric polynomials in an arbitrary number of variables supported in an arbitrary zonotope given by a set of generating vectors.

Vorheriger Artikel Symbolic Algorithm for Finding Zeros of a System of Holomorphic Functions

Nächster Artikel Symbolic-Numerical Implementation of the Galerkin Method for Approximate Solution of the Waveguide Diffraction Problem

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Abramov, S.A., Ryabenko, A.A., and Khmelnov, D.E., Laurent, rational, and hypergeometric solutions of linear q-difference systems of arbitrary order with polynomial coefficients, Program. Comput. Software, 2018, vol. 44, pp. 120–130.MathSciNetCrossRefMATH

Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A.V., Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, 1994.CrossRefMATH

Viro, O., What is an amoeba?, Not. AMS, 2002, vol. 49, no. 8, pp. 916–917.

Cherkis, S.A. and Ward, R.S., Moduli of monopole walls and amoebas, J. High Energy Phys., 2012, no. 5.

Fujimori, T., Nitta, M., Ohta, K., Sakai, N., and Yamazaki, M., Intersecting solitons, amoeba, and tropical geometry, Phys. Rev. D: Part., Fields, Gravitation, Cosmol., 2008, vol. 78, no. 10.

Kenyon, R., Okounkov, A., and Sheffield, S., Dimers and amoebae, Ann. Math., 2006, vol. 163, pp. 1019–1056.MathSciNetCrossRefMATH

Passare, M., Pochekutov, D., and Tsikh, A., Amoebas of complex hypersurfaces in statistical thermodynamics, Math., Phys., Anal., Geom., 2013, vol. 16, no. 1, pp. 89–108.MathSciNetCrossRefMATH

Zahabi, A., Quiver asymptotics and amoeba: Instantons on toric Calabi–Yau divisors, Phys. Rev. D, 2021, vol. 103, no. 8.

Maeda, T. and Nakatsu, T., Amoebas and instantons, Int. J. Modern Phys. A, 2007, vol. 22, no. 5, pp. 937–983.MathSciNetCrossRefMATH

10.

Mikhalkin, G., Real algebraic curves, the moment map and amoebas, Ann. Math., 2000, vol. 151, no. 2, pp. 309–326.MathSciNetCrossRefMATH

11.

Forsberg, M., Amoebas and Laurent series, Doctoral Thesis, Stockholm: Royal Institute of Technology (KTH), 1998.

12.

Leksell, M. and Komorowski, W., Amoeba program: Computing and visualizing amoebas for some complex-valued bivariate expressions. http://qrf.servequake.com/amoeba/AmoebaProgram.pdf.

13.

Rullgård, H., Topics in geometry, analysis, and inverse problems, Doctoral Thesis, Stockholm University, 2003. http://www.diva-portal.org/smash/get/diva2:190169/FULLTEXT01.pdf.

14.

Theobald, T., Computing amoebas, Exp. Math., 2002, vol. 11, no. 4, pp. 513–526.MathSciNetCrossRefMATH

15.

Timme, S., A package to compute amoebas in 2 and 3 variables. https://github.com/saschatimme/PolynomialAmoebas.jl.

16.

Theobald, T. and De Wolff, T., Approximating amoebas and coamoebas by sums of squares, Math. Comput., 2015, vol. 84, no. 291, pp. 455–473.MathSciNetCrossRefMATH

17.

Purbhoo, K., A nullstellensatz for amoebas, Duke Math. J., 2008, vol. 141, no. 3, pp. 407–445.MathSciNetCrossRefMATH

18.

Forsgård, J., Matusevich, L.F., Mehlhop, N., and De Wolff, T., Lopsided approximation of amoebas, Math. Comput., 2018, vol. 88, pp. 485–500.MathSciNetCrossRefMATH

19.

Anthony, E., Grant, S., Gritzmann, P., and Rojas, J.M., Polynomial-time amoeba neighborhood membership and faster localized solving, Math. Visualization, 2015, vol. 38, pp. 255–277.MathSciNetCrossRefMATH

20.

Bogdanov, D.V., Kytmanov, A.A., and Sadykov, T.M., Algorithmic computation of polynomial amoebas, Lect. Notes Comput. Sci. (including Lect. Notes Artif. Intell. and Lect. Notes Bioinf.), 2016, vol. 9890, pp. 87–100.

21.

Nisse, M. and Sadykov, T.M., Amoeba-shaped polyhedral complex of an algebraic hypersurface, J. Geom. Anal., 2019, vol. 29, no. 2, pp. 1356–1368.MathSciNetCrossRefMATH

22.

Bogdanov, D.V. and Sadykov, T.M., Hypergeometric polynomials are optimal, Math. Z, 2020, vol. 296, no. 1–2, pp. 373–390.MathSciNetCrossRefMATH

23.

Forsberg, M., Passare, M., and Tsikh, A.K., Laurent determinants and arrangements of hyperplane amoebas, Adv. Math., 2000, vol. 151, pp. 45–70.MathSciNetCrossRefMATH

24.

Klausen, R.P., Kinematic singularities of Feynman integrals and principal A-determinants, J. High Energy Phys., 2022, no. 2.

Titel: Computing the Connected Components of the Complement to the Amoeba of a Polynomial in Several Complex Variables
verfasst von: T. A. Zhukov
T. M. Sadykov
Publikationsdatum: 01.04.2023
Verlag: Pleiades Publishing
Erschienen in: Programming and Computer Software / Ausgabe 2/2023
Print ISSN: 0361-7688
Elektronische ISSN: 1608-3261
DOI: https://doi.org/10.1134/S0361768823020159

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 2/2023

Computer Algebra Calculations in Supersymmetric Electrodynamics

Development of Algorithms and Software for Modeling Controlled Dynamic Systems Using Symbolic Computations and Stochastic Methods

Symbolic Algorithm for Finding Zeros of a System of Holomorphic Functions

Symbolic-Numerical Implementation of the Galerkin Method for Approximate Solution of the Waveguide Diffraction Problem

Solving the Cauchy Problem for a Three-dimensional Difference Equation in a Parallelepiped

Computing Level Lines of a Polynomial on the Plane

Premium Partner