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SUMMARY:Erhard Aichinger (JKU Linz\, Austria)
DTSTART:20210216T200000Z
DTEND:20210216T210000Z
DTSTAMP:20260423T004137Z
UID:PALS/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PALS/1/">The
  degree as a measure of complexity of functions on a universal algebra</a>
 \nby Erhard Aichinger (JKU Linz\, Austria) as part of PALS Panglobal Algeb
 ra and Logic Seminar\n\n\nAbstract\nThe degree of a function $f$ between t
 wo abelian groups has been\ndefined as the smallest natural number $d$ suc
 h that\n$f$ vanishes after $d+1$ applications\nof any of the difference op
 erators $\\Delta_a$ defined by\n$\\Delta_a * f \\\,\\\, (x) = f(x+a) - f(x
 )$.\nFunctions of finite degree have also been called\ngeneralized polynom
 ials or solutions to Frechet's functional\n  equations. A pivotal result b
 y A. Leibman (2002) is that $\\deg (f \\circ g) \\le \\deg(f) \\cdot\n\\de
 g (g)$.\nWe show how results on the degree can be used\n(i) to get lower b
 ounds on the number of solutions of equations\, and\n(ii) to connect nilpo
 tency and supernilpotency.\nThis leads to generalizations of the Chevalley
 -Warning Theorems\nto abelian groups\, a group version of the Ax-Katz Theo
 rem on\nthe number of zeros of polynomial functions\, and a computable\n$f
 $ such that all finite $k$-nilpotent algebras of prime power order\nin con
 gruence modular varieties are $f(k\, .)$-supernilpotent.\n
LOCATION:https://researchseminars.org/talk/PALS/1/
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