The degree as a measure of complexity of functions on a universal algebra

Description

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.

Period	16 Feb 2021
Event title	PALS - Panglobal Algebra and Logic Seminar
Event type	Other
Location	AustriaShow on map