# Gibbs Manifolds

Section 3 features a symbolic implicitization algorithm for the Gibbs variety of a given LSSM defined over \(\mathbb{Q}\). We implemented this algorithm in Julia using the Oscar computer algebra package. Our code can be downloaded here: `GibbsCode.zip`

. An implementation of the algorithm can be found in `symbolic_implicitization.jl`

inside this archive.

Example 3.4 illustrates how numerical methods can be used to implicitize Gibbs varieties that are infeasible for symbolic computation: the defining equation is found numerically for the Gibbs variety of the following LSSM of Hankel matrices:

Julia code for this example can be downloaded here: `numerical_implicitization.jl`

(this file is also part of `GibbsCode.zip`

). The resulting equation has 853 terms and can be found here: `hankel4.txt`

.

Project page created: 25/11/2022.

Project contributors: Dmitrii Pavlov, Bernd Sturmfels, Simon Telen.

Corresponding author of this page: Dmitrii Pavlov, pavlov@mis.mpg.de.

Software used: Julia (Version 1.8.3), Oscar (Version 0.11.0).

System setup used: MacBook Pro with macOS Monterey 12.6, Processor 2,8 GHz Intel Core i7, Memory 16 GB 2133 MHz LPDDR3, Graphics Intel HD Graphics 630 1536 MB.

Last updated 29/11/22.