A remark on the composition of polynomial functions over algebraically closed fields

Description

In 1969, M.\ D.\ Fried and R.\ E.\ MacRae proved that for univariate polynomials $p,q, f, g \in \mathbb{K}[t]$ ($\mathbb{K}$ a field) with $p,q$ nonconstant, $p(x)-q(y)$ divides $f(x)-g(y)$ in $\mathbb{K}[x,y]$ if and only if there is $h \in \mathbb{K}[t]$ such that $f=h(p(t))$ and $g=h(q(t))$. In 1995, F.\ Binder and the author provided short algebraic proofs of this theorem, and J.\ Schicho gave a proof from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In this talk, we give an algebraic proof of one of these generalizations. % % The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions $f,g$ from $\mathbb{C}$ to $\mathbb{C}$: if both the composition $f \circ g$ and $g$ are polynomial functions, then $f$ has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. % As an application, one obtains a generalization of a result by L.\ Carlitz from 1963 that describes those univariate polynomials over finite fields that induce injective functions on all of their extensions. Part of this research is joint work with S.\ Steinerberger (Bonn, Germany).

Period	05 Feb 2011
Event title	AAA81 - 81. Arbeitstagung Allgemeine Algebra
Event type	Conference
Location	AustriaShow on map