A remark on the composition of polynomial functions over algebraically closed fields
Sprache des Vortragstitels:
Englisch
Original Tagungtitel:
AAA81 - 81. Arbeitstagung Allgemeine Algebra
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
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.
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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.
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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).
Forschungs-