Approximate sum of squares decompositions for Adj₅ + k·Op₅ - λΔ₅ ∈ ISAut(F₅)

This is the dataset accompanying On property (T) for Aut(Fₙ) and SLₙ(ℤ) paper (https://arxiv.org/abs/1812.03456). See the appendix thereof and Section 4 of (Aut(F₅) has property (T)) for more details.

Content

1. 1812.03456-cf6dee7.zip contains a julia environment specification (Project.toml and Manifest.toml) as well as 1812.03456.jl script used for automatic certification and jupyter noteboks in ./notebooks directory.
2. SAutF5_r2.tar.xz contains the precomputed solutions for expressing Adj₅+2·Op₅-0.28Δ₅ and Adj₅+3·Op₅-1.4Δ₅ as sum of (hermitian) squares in the group ring of SAut(F₅). The contents of this  archive must be placed inside `1812.03456`directory from the zip file.

Preparation

The code needs to be run with julia-1.4.0 or higher (tested versions include also versions julia-1.5). In principle any version in  [1.4-2.0) should work due to the promise of forward compatibility.

While located in the main directory (1812.03456) you should run the following code in julias REPL console to instantiate the environment for computations:

using Pkg
Pkg.activate(".")
Pkg.instantiate()

(this needs to be done once per installation). Then the directory SAutF5_r2 (from the SAutF5_r2.tar.xz archive) needs to be placed in 1812.03456.

Replication: Jupyter notebook

A jupyter server may be launched then within the directory 1812.03456 by issuing from julia command-line (REPL) the following commands.

using Pkg
Pkg.activate(".")
using IJulia
notebook(dir=".")

During the first run the user may be asked for installation of Jupyter program (a server for running this notebook) within miniconda environment, which will happen automatically after confirmation. To execute the commands in the notebook, one needs to navigate to notebooks subdirectory of 1812.03456 and click either of the notebooks.

One can replicate the main computational results of the paper by executing all the cells in the Positivity of Adj_n + kOp_n in ISAut(F_n) notebook.

Replication: script

To verify that (Adj₅ + 3.0·Op₅) - 1.4·Δ₅ admits an approximate sum of squares decomposition run in 1812.03456 directory

julia --project=. --color=yes 1812.03456.jl -n 5 -k 3 -l 1.4

On a modern laptop computer this should finish in less than 2h.

At the end of computations you will see lines such as:

┌ Info: λ is certified to be >
└   λ_cert.lo = 1.3701131733828074
[ Info: i.e Adj_5 + 3.0·Op_5 - (1.3701131733828074)·Δ_5 ∈ Σ²₂ ISAut(F_5)

This means that  Adj₅ + 3.0·Op₅ - λΔ₅ is a sum of Hermitian squares of elements from ISAut(F₅) for every λ < 1.370....

A similar verification for Adj₅ + 2.0·Op₅ - 0.28·Δ₅ can be run by executing

julia --project=. --color=yes 1812.03456.jl -n 5 -k 2 -l 0.28

Generating the provided files

If you wish to produce the whole certificate on your own (including the generation of group ring and its multiplication table), delete all *.jld files from the SAutF5_r2 folder and run one of the above commands with the same (or different) parameters again. Note: To do this you need at least 16GB of RAM and spare 24h of your CPU.

This research was supported in part by National Science Center, Poland, grant 2017/26/D/ST1/00103.