Geodesic
Heights and totally $p$-adic numbers
Lukas Pottmeyer1
1 Fachbereich Mathematik Universit\"at Basel Spiegelgasse 1 4051 Basel, Switzerland
Acta Arithmetica, Tome 171 (2015) no. 3, pp. 277-291

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.
DOI : 10.4064/aa171-3-5
Keywords: study behavior canonical height functions widehat associated rational maps totally p adic fields particular prove there gap between zero smallest value widehat maximal totally p adic field map has least periodic point contained field application prove there infinite subset compositum number fields degree non linear polynomial nbsp answers question nbsp narkiewicz

Lukas Pottmeyer 1

1 Fachbereich Mathematik Universit\"at Basel Spiegelgasse 1 4051 Basel, Switzerland 
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Lukas Pottmeyer. Heights and totally $p$-adic numbers. Acta Arithmetica, Tome 171 (2015) no. 3, pp. 277-291. doi: 10.4064/aa171-3-5

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