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

  1. Philippe Gimenez 1
  2. Andrés Sáez Schwedt 2
  3. Tomás Sánchez Giralda 3
  1. 1 IMUVA-Mathematics Research Institute, Universidad de Valladolid
  2. 2 Universidad de León
    info

    Universidad de León

    León, España

    ROR https://ror.org/02tzt0b78

  3. 3 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

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Libro:
EACA 2022: XVII Encuentro de Álgebra Computacional y Aplicaciones
  1. Galindo Pastor, Carlos (coord.)
  2. Gimenez, Philippe (coord.)
  3. Hernando Carrillo, Fernando (coord.)
  4. Monserrat Delpalillo, Francisco José (coord.)
  5. Moyano-Fernández, Julio José (coord.)

Editorial: Servei de Comunicació i Publicacions ; Universitat Jaume I

ISBN: 978-84-19647-46-7

Año de publicación: 2023

Páginas: 99-102

Congreso: Encuentro de Álgebra Computacional y Aplicaciones (17. 2022. Castelló de la Plana)

Tipo: Aportación congreso

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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.

Fuente de los datos: Dialnet