Abstract
Algorithms and software for two new methods of solving polynomial equations based on constructing a convex polygon are described. The first method approximately solves the equations using the Hadamard polygon. The second method makes it possible to find branches of an algebraic curve in the vicinity of its singular points and in the vicinity of infinity using the Newton polygon and sketch real algebraic curves on the plane. The corresponding geometries and computer algebra algorithms for analyzing arbitrarily complicated cases are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bruno, A.D., Algorithms for solving an algebraic equation, Program. Comput. Software, 2018, vol. 44, no. 6, pp. 533–545. https://doi.org/10.1134/S0361768819100013
Bruno, A.D., Power Geometry in Algebraic and Differential Equations (Moscow: Fizmatlit, 1998; Amsterdam: Elsevier, 2000).
Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert, Berlin: 1926. Development of Mathematics in the 19th Century, Berlin: Springer, 1979.
Efimov, N.V., Higher Geometry, Moscow: Mir, 1980.
Kostrikin A.I. and Manin Y.I., Linear Algebra and Geometry, New York: Gordon & Breach, 1997.
Veblen O. and Young J. W., Projective Geometry, Vol. 1 New York: Blaisdell, 1938.
Shafarevich, I.R., Basic Algebraic Geometry 1. Varieties in Projective Space, Heidelberg: Springer, 2013.
Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989.
Bruno, A.D. and Batkhin, A.B., Resolution of an algebraic singularity by power geometry algorithms, Program. Comput. Software, 2012, vol. 38, no. 2, pp. 57–72. https://doi.org/10.1134/S036176881202003X
Bruno, A.D., Asymptotic behaviour and expansions of solutions of an ordinary differential equation, Russ. Math. Surveys, 2004, vol. 59, no. 3, pp. 429–480.
Bruno, A.D., and Shadrina, T.V., Axisymmetric boundary layer on a needle, Trans. Mosc. Math. Soc., 2007, vol. 68, pp. 201–259.
Bruno, A.D. and Goryuchkina, I.V., Asymptotic expansions of solutions to the sixths Painlevé equation, Trans. Mosc. Math. Soc., 2010, vol. 71, pp. 1–104.
Bruno, A.D., Expansion of solutions to an ordinary differential equation into transseries, Dokl. Math, 2019, vol. 99, no. 1, pp. 36–39. https://doi.org/10.1134/S1064562419010113
Gallier, J., Geometric Methods and Applications. For Computer Science and Engineering. New York: Springer, 2011.
Barber C.B. Dobkin D.P., and Huhdanpaa H.T., The Quickhull algorithm for convex hulls, ACM Trans. Math. Software, 1996, vol. 22, no. 4, pp. 469–483.
Batkhin, A.B., Bruno, A.D., and Varin, V.P., Stability sets of multiparameter Hamiltonian systems, J. Appl. Math. Mech. 2012, vol. 76, no. 1, pp. 56–92. https://doi.org/10.1016/j.jappmathmech.2012.03.006
Thompson, I., Understanding Maple, Cambridge University Press, 2016.
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.1.1), 2020. https://doi.org/10.5281/zenodo.4066866. https://www.sagemath.org.
Fukuda K. cdd, cddplus and cddlib homepage. McGill University, Montreal, Canada, 2002. http://www.cs.mcgill.ca/~fukuda/software/cdd_home/ cdd.html
Bagnara R., Hill P.M., and Zaffanella E., The Parma polyhedra library: toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems, Sci. Comput. Program., 2008, vol. 72, nos. 1–2, pp. 3–21.
Meurer, A., Smith, C.P., Paprocki, M., et al. SymPy: Symbolic computing in Python, PeerJ Comput. Sci., 2017, vol. 3, p. e103. https://doi.org/10.7717/peerj-cs.103
Bruce, King R., Beyond the Quartic Equation. Boston: Birkhäser, 1996.
Umemura, H., Resolution of algebraic equations by theta constants, in D. Mumford, Tata Lectures on Theta II. Jacobian Theta Functions and Differential Equations (Birkhäuser, Boston, 1984), pp. 261–272.
Hadamard, J., Etude sur les propriétés des fonctions entières et en particulier d’une fonction considéro par Riemann, J. Math. Pures Appl. \({{4}^{e}}\) Sér. 1893, vol. 9, pp. 171–216.
Kurosh, A.G. Higher Algebra, Moscow, Mir, 1980.
Kalinina, E.A. and Uteshev, A.Yu., Elimination Theory, St. Petersburg: Naucho-Issledovatel’skii Inst. Khimii, St. Petersburg Univ., 2002 [in Russian].
Von zur Gathen J. and Lücking, T. Subresultants revisited, Theor. Comput. Sci., 2003, vol. 297, pp. 199–239. https://doi.org/10.1016/S0304-3975(02)00639-4
Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 67–76. https://doi.org/10.1134/S0361768816020031
Akritas, A.G., Elements of Computer Algebra with Applications, New York: Wiley, 1989.
Puiseux, V., Recherches sur les fonctions algébriques, J. Math. Pures Appl., Sér. 1, 1850, vol. 15, pp. 365–480.
Goursat, E., Course of Mathematical Analysis, New York: Dover, 1959.
Hoeij M., Rational parametrizations of algebraic curves using a canonical divisor, J. Symb. Comput., 1997, vol. 23, pp. 209–227.
Wolfram, S., The Mathematica Book, Wolfram Media, 2003.
Cox D., Little J., and O’Shea D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Heidelberg: Springer, 2015, 4th ed.
Zobova, I., Geometric methods of solving polynomial equations, Materialy XIX Mezhd. Konf., Shkol’nikov Kolmogorovskie Chteniya (Abstracts of the XIX Conf. of Students of the Advanced Educational Scientific Center—Kolmogorov Boarding School of Moscow State University) 2019, Matematika, Moscow: Shkola-Internat im. Kolmogorova, 2019, pp. 15–16.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by A. Klimontovich
Rights and permissions
About this article
Cite this article
Bruno, A.D., Batkhin, A.B. Algorithms and Programs for Calculating the Roots of Polynomial of One or Two Variables. Program Comput Soft 47, 353–373 (2021). https://doi.org/10.1134/S0361768821050042
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768821050042