Work

Computing H-equations with 2-by-2 integral matrices

Year: 2025

Type: article

Abstract: We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral invertible matrices: “given h 1 , … , h r ; g ∈ PSL 2 ( Z ) , decide whether g is algebraic over the subgroup H = 〈 h 1 , … , h r 〉 ⩽ PSL 2 ( Z ) (i.e., whether there exist a non-trivial H -equation w ( x ) ∈ H ⁎ 〈 x 〉 such that w ( g ) = 1 ) and, in the affirmative case, compute finitely many such H -equations w 1 ( x ) , … , w s ( x ) ∈ H ⁎ 〈 x 〉 further satisfying that any w ( x ) ∈ H ⁎ 〈 x 〉 with w ( g ) = 1 is a product of conjugates of w 1 ( x ) , … , w s ( x ) ”. The same problem for square matrices of size 4 and bigger is unsolvable.

Source: Linear Algebra and its Applications

DOI: 10.1016/j.laa.2025.06.007

Authors: Gemma Bastardas, Enric Ventura

Institution: Universitat Politècnica de Catalunya

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,230

Funders: Federación Española de Enfermedades Raras, Agencia Estatal de Investigación