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A new invariant for difference fields

Banff International Research Station for Mathematical Innovation and Discovery

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 $dd(a/K) = lim [K(f^k(a),a):K(a)]^{1/k}$, 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 $Aut(K(a)^{alg}/K)$ as a locally compact group, this result is connected to results of Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results.

Mathematics; Group theory and generalizations; Topological groups, lie groups; Algebraic structures

video/mp4

Author affiliation: CNRS - Ecole Normale Supérieure

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61675

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/