On Taylor series of zeros with general base function

Abstract

We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux series. We prove an integrality result about these coefficients which implies and generalizes the integrality result of Sturmfels ("Solving algebraic equations in terms of $\mathcal{A}$-hypergeometric series". Discrete Math. 210 (2000) pp. 171-181). We also prove a transformation rule for a special case of these Taylor series.