Replication code for https://arxiv.org/abs/2207.02783

Replication details for [2207.02783](https://arxiv.org/abs/2207.02783)

For the computations we used julia in version `1.8.3` but in principle any later version should work.

Setting up the environment

First, unpack the file "LowCohomologySOS.zip" and run Julia in the terminal in `LowCohomologySOS` folder

julia --project=.

Next, to set up the proper environment for the replication run in julia REPL

julia> using Pkg; Pkg.instantiate()

This command installs and precompiles, if needed, all the necessary dependences, so it may take a while. Note that this step needs to be executed only once per installation.

Running actual replication

We wish to prove that for for the Steinberg presentation of \(\text{SL}_3(\mathbb{Z})\) on six generators (as defined in Section 2 of [2207.02783](https://arxiv.org/abs/2207.02783)) \(\Delta_1-\lambda I_6\) is a sum of squares for some \(\lambda\geq 0.32\). We provide a script which performs the necessary optimization to find such sum of squares decomposition.

As before the following command needs to be executed in the terminal in `LowCohomologySOS` folder:

julia --project=. ./scripts/SL_3_Z_Delta_1.jl

The running time of the script will be approximately `2` hours on a standard laptop computer.

Alternatively, you can run the script which uses the precomputed solution (stored in the file "sl_3_z_precomputed.sjl") and provides rigorous proof (certification) of the result. In order to do this, the following command needs to be executed in the terminal in `LowCohomologySOS` folder:

julia --project=. ./scripts/SL_3_Z_Delta_1_precomputed.jl

The running time of the script will be approximately `2` minutes on a standard laptop computer.