A new invariant for difference fields

Zoe Chatzidakis (CNRS - ENS)

19-Apr-2021, 06:30-07:30 (3 years ago)

Abstract: If $(K,f)$ is a difference field, and a is a finite tuple in some difference field extending $K$, and such that $f(a)$ in $K(a)^{alg}$, then we define $dd(a/K)=\mathop{lim}[K(f^k(a),a):K(a)]^{1/k}$, the distant degree of $a$ over $K$. This is an invariant of the difference field extension $K(a)^{alg}/K$. We show that there is some $b$ in the difference field generated by $a$ over $K$, which is equi-algebraic with $a$ over $K$, and such that $dd(a/K)=[K(f(b),b):K(b)]$, i.e.: for every $k>0$, $f(b) \in K(b,f^k(b))$.

Viewing $\mathop{Aut}(K(a)^{alg}/K)$ as a locally compact group, this result is connected to results of Goerge Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results. (Joint with E. Hrushovski)

group theory

Audience: researchers in the discipline


Symmetry in Newcastle

Organizer: Michal Ferov*
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