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

Abstract: 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 Frechet'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 (i) to get lower bounds on the number of solutions of equations, and (ii) 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.

category theorylogic

Audience: researchers in the topic

PALS Panglobal Algebra and Logic Seminar

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