Document Type : Review Article

Author

• André G. LEROY

Université d' Artois, Faculté des Sciences

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 ($\si,\de$)-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 ($\si, \delta$)-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.

Keywords

• skew polynomial rings
• polynomial maps
• differential equations