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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
Published electronically: May 17, 2019
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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

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