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The aim is to provide an overview of pseudo-linear maps, detailing their historical background and properties compared to linear maps.

April 13, 2026Open Access

Pseudo-linear maps, an overview

Key Points

The aim is to provide an overview of pseudo-linear maps, detailing their historical background and properties compared to linear maps.
Survey historical context of pseudo-linear maps since 1937
Introduce and define ($ si, de$)-pseudo-linear transformations
Examine analogies between linear maps and ($ si, de$)-PLT's
Present examples and applications in evaluating polynomial classes
Established connections between pseudo-linear maps and evaluation of skew polynomials
Demonstrated relevance of Ore polynomials to sequences of pseudo-linear maps
Identified new methods to address an open question through these transformations

Abstract

In this survey we give a brief account of the history of pseudo-linear maps since their introduction by N. Jacobson in 1937. The (, ) -pseudo-linear transformations are introduced via the modules over a skew polynomial ring. Many classical properties of linear maps are shown to have analogues in the setting of (, ) -PLT's. The relations between these maps and the evaluation of skew polynomials is particularly emphasized. It is shown, in particular, how they are useful while evaluation inside a conjugacy class is considered. The evaluation of Ore polynomials with more then one variables is shown to be related to sequence of pseudo-linear maps. We also briefly present a recent application that allows to easily answer an open question. Many examples are presented all along the text.

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Pseudo-linear maps, an overview

Key Points

Abstract

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