The degree as a measure of complexity of functions on a universal algebra
Sprache des Vortragstitels:
Englisch
Original Tagungtitel:
PALS - Panglobal Algebra and Logic Seminar
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
The \emph{degree} of a function $f$ between two abelian groups has been
defined as the smallest natural number $d$ such that
$f$ vanishes after $d+1$ applications
of any of the difference operators $\Delta_a$ defined by
$\Delta_a * f \,\, (x) = f(x+a) - f(x)$.
Functions of finite degree have also been called
\emph{generalized polynomials} or \emph{solutions to Fr\'echet's functional
equations}. A pivotal result by A.\ Leibman (2002) is that $\deg (f \circ g) \le \deg(f) \cdot
\deg (g)$.
We show how results on the degree can be used
\begin{itemize}
\item to get lower bounds on the number of solutions of equations, and
\item to connect nilpotency and supernilpotency.
\end{itemize}
This leads to generalizations of the Chevalley-Warning Theorems
to abelian groups, a group version of the Ax-Katz Theorem on
the number of zeros of polynomial functions, and a computable
$f$ such that all finite $k$-nilpotent algebras of prime power order
in congruence modular varieties are $f(k, .)$-supernilpotent.