Abstract

A lot of research in on polynomial maps is induced by the notion that they are a natural extension of linear maps. Of course, we have the notorious Jacobian Conjecture, induced by a triviality for linear maps: if $F$ is a polynomial map, then $det(Jac(F))$=nonzero constant implies $F$ is invertible. Well, what is the most powerful theorem in linear algebra? In my opinion: the Cayley-Hamilton theorem. Note, the determinant of a linear map occurs as the last coefficient in the characteristic polynomial, which may indicate a link with the Jacobian Conjecture. So why has there not been an attempt to generalize the Cayley-Hamilton theorem to polynomial maps? Well, a simple answer is: that is not possible. However: take a look at the map $F\; =\; (X\; +\; Y2,\; Y\; )$ on $C2$ (i.e. $(x,\; y)\; \to \; (x\; +\; y2,\; y)$). Then you can see that $F2-\; 2F\; +\; I\; =\; 0$. So, $F$ is a zero of $T2-\; 2T\; +\; 1$. The set of polynomial maps that are zero of a polynomial $P(T)$ we baptized the Locally Finite polynomial maps (short LF).(1) Though this set was originally discovered by the speaker as a fun object, it slowly became clear that these maps have quite some potential: among others, they may hold the key to a better understanding of the group of invertible polynomial automorphisms! In this talk I will introduce LF maps, give an actual extension of the Cayley-Hamilton theorem to LF maps, and pose some very puzzling questions.

(1)The LF maps is a subset of the Dynamically trivial polynomial maps, which may make them, at least in name, less interesting to dynamical systems experts - but if they are so trivial, then please solve my questions!