On computing local monodromy and the numerical local irreducible decomposition

TL;DR

This paper introduces an algorithm for computing local monodromy and local irreducible decompositions of algebraic varieties using numerical methods, extending the analysis beyond local neighborhoods.

Contribution

It presents a novel algorithm for numerical local irreducible decomposition based on local monodromy computation, with theoretical characterization and software implementation.

Findings

Algorithm successfully computes local monodromy actions.

Numerical decompositions align with theoretical expectations.

Implementation demonstrated on multiple example varieties.

Abstract

Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The key requirement for obtaining a numerical local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well-defined on any small enough neighborhood. We characterize some of the behavior of local monodromy action of linear projection maps under analytic continuation, allowing computations to be performed beyond a local neighborhood. With this characterization, we present an algorithm to compute the local monodromy action and corresponding numerical local irreducible decomposition for algebraic varieties. The results are illustrated using several examples facilitated by an implementation in an open source software package.