Convergence and many-valuedness of hensel seriesnear the expansion point

Abstract

Hensel series is an expansion of multivariate algebraic function at a singular point, computed from the defining polynomial by the Hensel construction. The Hensel series is well-structured and tractable, hence it seems to be useful in various applications. In SNC'07, the present authors reported the following interesting properties of Hensel series, which were found numerically. 1) The convergence and the divergence domains co-exist in any small neighborhood of the expansion point. 2) If we trace a Hensel series by passing a divergence domain, the series may jump from a branch to another branch of the original algebraic function. In this paper, we clarify these properties theoretically and derive stronger properties.