On Taylor series of zeros with general base function

TL;DR

This paper derives a general formula for the Taylor series coefficients of zeros of sums involving complex-exponent polynomials and holomorphic base functions, extending previous integrality results and providing transformation rules.

Contribution

It introduces a new, more general Taylor series formula for zeros of complex functions, extending prior integrality results and including a transformation rule for special cases.

Findings

Established a formula for Taylor series coefficients of zeros involving complex-exponent polynomials and holomorphic functions.

Proved an integrality property of these coefficients, generalizing Sturmfels' earlier results.

Derived a transformation rule for specific cases of these Taylor series.

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.