The finiteness conjecture for $3 \times 3$ binary matrices

TL;DR

This paper enhances the invariant polytope algorithm with mixed computations and input augmentation, enabling the proof of the finiteness conjecture for all pairs of binary 3x3 matrices and sign 2x2 matrices.

Contribution

It introduces a modified invariant polytope algorithm combining numeric and symbolic methods to solve the finiteness conjecture for specific matrix classes.

Findings

Proved the finiteness conjecture for all pairs of binary 3x3 matrices.

Proved the finiteness conjecture for all sign 2x2 matrices.

Enhanced the algorithm's applicability to broader matrix families.

Abstract

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many applications in problems of functional analysis, approximation theory, combinatorics, etc. In this paper we propose a modification of the invariant polytope algorithm enlarging the class of problems to which it is applicable. Precisely, we introduce mixed numeric and symbolic computations. A further minor modification of augmenting the input set with additional matrices speeds up the algorithm in certain cases. With this modifications we are able to automatically prove the finiteness conjecture for all pairs of binary $3\times 3$ matrices and sign $2\times 2$ matrices.