Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 4

Special Issue: In Honor of Prof. Gert-Martin Greuel’s 75th Birthday

Guest Editors: Igor Burban, Stanislaw Janeczko, Gerhard Pfister, Stephen S.T. Yau, and Huaiqing Zuo

A note on finite determinacy of matrices

Pages: 1115 – 1121

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n4.a10


Thuy Huong Pham (Department of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Vietnam)

Pedro Macias Marques (Departamento de Matemática, CIMA, IIFA, Universidade de Évora, Portugal)


In this note, we give a necessary and sufficient condition for a matrix $A \in M_{2,2}$ to be finitely $G$-determined, where $M_{2,2}$ is the ring of $2 \times 2$ matrices whose entries are formal power series over an infinite field, and $G$ is a group acting on $M_{2,2}$ by change of coordinates together with multiplication by invertible matrices from both sides.


equivalence of matrices, finite determinacy, group actions in positive characteristic, tangent image to orbit

2010 Mathematics Subject Classification

13A50, 13F25, 14B05

The first author was partially supported by the European Union’s Erasmus+ programme. She would also like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for its support and hospitality.

The second author was partially supported by CIMA – Centro de Investigação em Matemática e Aplicações, Universidade de Évora, project PEst-OE/MAT/UI0117/2014 (Fundação para a Ciência e Tecnologia).

Received 4 January 2019

Accepted 24 October 2019

Published 13 November 2020