A note on ﬁnite determinacy of matrices

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.

Keywords

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

2010 Mathematics Subject Classification

13A50, 13F25, 14B05

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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