Erhard Aichinger (JKU Linz, Austria), The degree as a measure of complexity of functions on a universal algebra

Tue, 16 Feb 2021, 1 pm MST

The 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 generalized polynomials or solutions to Fréchet's functional equations. A pivotal result by A. Leibman (2002) is that $\mathrm{deg} (f \circ g) \le \mathrm{deg}(f) \cdot\mathrm{deg} (g)$. We show how results on the degree can be used

  • to get lower bounds on the number of solutions of equations, and
  • to connect nilpotency and supernilpotency.
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.

[slides] [video]