On holomorphicity of Hartogs series satisfying algebraic relations

TL;DR

This paper proves that a formal power series with holomorphic coefficients, satisfying certain convergence and algebraic conditions, must be holomorphic on the product domain, extending understanding of holomorphicity in algebraic series.

Contribution

It establishes a new theorem linking algebraic relations and holomorphicity for Hartogs series in several complex variables.

Findings

Series defines a holomorphic function on the product space

Condition (C2) is shown to be essential through an example

Provides a criterion for holomorphicity based on algebraic and convergence conditions

Abstract

We consider a formal power series in one variable whose coefficients are holomorphic functions in a given multidimensional complex domain. Assume the following two conditions on the series. (C1) The restriction of the series at each point of a dense subset of the domain converges in an open disk of a fixed radius. (C2) The series is algebraic over the ring of holomophic functions on the direct product space of the domain and the disk. The main theorem of the present note is that the series defines a holomorphic function on the direct product space. We also give an example where the condition (C2) is essentially necessary.