Vol. 11, No. 1, 2018

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ISSN: 1944-4184 (e-only)
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Normal forms of endomorphism-valued power series

Christopher Keane and Szilárd Szabó

Vol. 11 (2018), No. 1, 81–94
DOI: 10.2140/involve.2018.11.81
Abstract

We show for n,k 1, and an n-dimensional complex vector space V that if an element A End(V )[[z]] has constant term similar to a Jordan block, then there exists a polynomial gauge transformation g such that the first k coefficients of gAg1 have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first nk coefficients of the Puiseux series expansion of the eigenvalues of A and the entries of the first k coefficients of gAg1.

Keywords
normal form, endomorphism, formal power series, Puiseux series
Mathematical Subject Classification 2010
Primary: 15A18, 15A21, 15A54
Secondary: 05E40
Milestones
Received: 17 July 2016
Revised: 31 August 2016
Accepted: 17 October 2016
Published: 17 July 2017

Communicated by Kenneth S. Berenhaut
Authors
Christopher Keane
Department of Mathematics
Reed College
3203 SE Woodstock Blvd
Portland, OR 97202
United States
Szilárd Szabó
Department of Mathematics
Budapest University of Technology and Economics
Egry J. u. 1, H ep.
Budapest
1111
Hungary