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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Closed-form solutions to irreducible Newton-Puiseux equations by Lagrange inversion formula and diagonalization on polynomial sequences of binomial-type
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by Soowhan Yoon PDF
Proc. Amer. Math. Soc. 147 (2019), 4585-4596 Request permission

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.
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Additional Information
  • Soowhan Yoon
  • Affiliation: Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822
  • Email: yoon@math.hawaii.edu
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4585-4596
  • MSC (2010): Primary 05A40, 14H20, 30C15
  • DOI: https://doi.org/10.1090/proc/14580
  • MathSciNet review: 4011496