LocalMonodromy.jl

Version

1.0.1

Copyright (C) 2024 Parker Edwards

Installation

For global installation from Julia in package mode (press ] to enter package mode, backspace to exit): add https://github.com/P-Edwards/LocalMonodromy.jl.git

To install as a standalone local package: Clone this repository to a directory. Then, from Julia in package mode: activate <path/to/cloned/copy/of/project/root> followed by instantiate.

Basic usage

To use multiple cores, run export JULIA_NUM_THREADS=<number_to_use> before starting Julia. Examples are available at https://doi.org/10.5281/zenodo.14532555.

This package primarily exposes the following function:

NumericalLocalIrreducibleDecomposition(system,point,vars=variables(system);max_tries=3)

Where

• system is a vector of polynomial expressions
• point is a Vector{Complex} at which to compute the numerical local irreducible decomposition.
• vars is a Vector{ModelKit.Variable} specifying the variable order.
• max_tries is an Int specifying how many times to retry if numerical failures are detected.

The output is a NumericalLocalIrreducibleDecomposition, a type which is overviewed below.

Example: Compute a numerical local irreducible decomposition.

julia> using LocalMonodromy, HomotopyContinuation
julia> @var x y z
julia> result = NumericalLocalIrreducibleDecomposition([x^2+y^2+z^2],[0;0;0])
 Numerical local irreducible decomposition with 1 components
• 1 component(s) of dimension 2.

 degree table of components:
╭───────────┬───────────────────────╮
│ dimension │ degrees of components │
├───────────┼───────────────────────┤
│     2     │           2           │
╰───────────┴───────────────────────╯

The output result is of type NumericalLocalIrreducibleDecomposition which has one field:

• result.Local_Witness_Sets is a Dict(Int,Vector{LocalWitnessSets}), which contains 1 vector of LocalWitnessSet for each dimension. Currently, only pure dimension is supported.

Example

julia> result.Local_Witness_Sets[2]
1-element Vector{LocalWitnessSet}

Each LocalWitnessSet has the following fields:

• F the polynomial system
• L a LinearSubspace for slicing
• R a vector of localized witness points in V(F)\cap R
• point the point at which the input was localized

Paper

This package implements the algorithm detailed in "On computing local monodromy and the numerical local irreducible decomposition" by Parker B. Edwards and Jonathan D. Haunstein available at https://doi.org/10.48550/arXiv.2412.17952.

License

This repository is distributed under GPLv3.