Computation of canonical forms for single input linear systems over Hermite rings

• Philippe Gimenez ^[1] ; Andrés Sáez Schwedt ^[2] ; Tomás Sánchez Giralda ^[1]
  1. [1] Universidad de Valladolid

     Universidad de Valladolid

     Valladolid, España

     
  2. [2] Universidad de León

     Universidad de León

     León, España

     
• Localización: EACA 2022: XVII Encuentro de Álgebra Computacional y Aplicaciones / coord. por Carlos Galindo Pastor, Philippe Giménez, Fernando Javier Hernando Carrillo, F. Monserrat Delpalillo, Julio José Moyano Fernández, 2023, ISBN 978-84-19647-46-7, págs. 99-102
• Idioma: inglés
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  • Texto Completo Libro
• Resumen

  • Let R be a commutative ring with the property that unimodular vectors can be completed to invertible matrices. Such a ring is called Hermite in the sense of Lam [4].In this note we construct a canonical form for matrix pairs (A, b), where A ∈ Rn×n and b ∈ Rn×1, under the feedback equivalence relation: (A, b) and (A, b) are equivalent if and only if A = P AP −1 + P bK and b = P b, for some matrices P ∈ GLn(R) and K ∈ R1×n.When R is a principal ideal domain, the canonical form is easily computed by using standard Hermite and Smith normal forms. When R is a polynomial ring K[x1, . . . , xt ], with K a field, the previously obtained canonical form remains valid, and can be determined by means of effective calculations. Our procedure consists mainly of elementary operations, combined with an adaptation of the currently available algorithms to solve theunimodular completion problem, used in the context of the Quillen-Suslin’s theorem that solves Serre’s conjecture.