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

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.