Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Closed-form solutions to irreducible Newton-Puiseux equations by Lagrange inversion formula and diagonalization on polynomial sequences of binomial-type


Author: Soowhan Yoon
Journal: Proc. Amer. Math. Soc. 147 (2019), 4585-4596
MSC (2010): Primary 05A40, 14H20, 30C15
DOI: https://doi.org/10.1090/proc/14580
Published electronically: May 17, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent article published in 2017, Barroso, Pérez, and Popescu-Pampu employ the Lagrange inversion formula to solve certain Newton-Puiseux equations when the solutions to the inverse problems are given. More precisely, for an irreducible $ f(x,y)\in K[[x,y]]$ over an algebraically closed field $ K$ of characteristic zero, they calculate the coefficients of $ \eta (x^{1/n})$ which would meet $ f(x,\eta (x^{1/n}))=0$ in terms of the coefficients of $ \xi (y^{1/m})$ that satisfy $ f(\xi (y^{1/m}),y)=0$. This article will present an alternative approach to solving the problem using diagonalizations on polynomial sequences of binomial-type. Along the way, a close relationship between binomial-type sequences and the Lagrange inversion formula will be observed. In addition, it will extend the result to give the coefficients of $ \eta (x^{1/n})$ directly in terms of the coefficients of $ f(x,y)$. As an application, an infinite series formula for the roots of complex polynomials will be obtained together with a sufficient condition for its convergence.


References [Enhancements On Off] (What's this?)

  • [1] Shreeram Shankar Abhyankar, Inversion and invariance of characteristic pairs, Amer. J. Math. 89 (1967), 363-372. MR 0220732, https://doi.org/10.2307/2373126
  • [2] Evelia Rosa García Barroso, Pedro Daniel González Pérez, and Patrick Popescu-Pampu, Variations on inversion theorems for Newton-Puiseux series, Math. Ann. 368 (2017), no. 3-4, 1359–1397. MR 3673657, https://doi.org/10.1007/s00208-016-1503-1
  • [3] E. T. Bell, The History of Blissard’s Symbolic Method, with a Sketch of its Inventor’s Life, Amer. Math. Monthly 45 (1938), no. 7, 414–421. MR 1524334, https://doi.org/10.2307/2304144
  • [4] Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. MR 1782072
  • [5] G. Halphen, Sur une série de courbes analogues aux développées, Journal de maths. pures et appliquées (de Liouville) 3e série, tome 2, pp. 87-144 (1876).
  • [6] Joseph P. S. Kung, Gian-Carlo Rota, and Catherine H. Yan, Combinatorics: the Rota way, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2009. MR 2483561
  • [7] J.-L. Lagrange, Nouvelle méthode pour résoudre les Équations littérales par le moyen des séries, Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 24, pp. 251-326 (1770).
  • [8] Isaac Newton, La méthode des fluxions et des suites infinies, Traduit par M. de Buffon, Librairie Scientifique Albert Blanchard, Paris, 1966 (French). MR 0199076
  • [9] V. Puiseux, Recherches sur les fonctions algébriques, Journal de maths. pures et appliquées (de Liouville) 15, pp. 365-480 (1850).
  • [10] Steven Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185
  • [11] Gian-Carlo Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl. 42 (1973), 684-760. MR 0345826, https://doi.org/10.1016/0022-247X(73)90172-8
  • [12] Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260
  • [13] O. Stolz, Die Multiplicität der Schnittpunkte zweier algebraischen Curven, Math. Ann. 15 (1879), no. 1, 122–160 (German). MR 1510004, https://doi.org/10.1007/BF01444108
  • [14] Oscar Zariski, Studies in equisingularity. III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961-1023. MR 0237493, https://doi.org/10.2307/2373492

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05A40, 14H20, 30C15

Retrieve articles in all journals with MSC (2010): 05A40, 14H20, 30C15


Additional Information

Soowhan Yoon
Affiliation: Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822
Email: yoon@math.hawaii.edu

DOI: https://doi.org/10.1090/proc/14580
Received by editor(s): November 5, 2018
Received by editor(s) in revised form: February 1, 2019
Published electronically: May 17, 2019
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society