Heights and totally $p$-adic numbers
Acta Arithmetica 171 (2015), 277-291 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 $p$-ad...
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| creator | Pottmeyer, Lukas |
| description | Acta Arithmetica 171 (2015), 277-291 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_str_mv | 10.48550/arxiv.1504.04985 |
| format | Article |
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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
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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
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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.</abstract><doi>10.48550/arxiv.1504.04985</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Number Theory |
| title | Heights and totally $p$-adic numbers |
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