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We study the behavior of canonical height functions , 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 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.
@article{LukasPottmeyer2015, abstract = {We study the behavior of canonical height functions $ĥ_\{f\}$, 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 $ĥ_\{f\}$ 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.}, author = {Lukas Pottmeyer}, journal = {Acta Arithmetica}, keywords = {height bounds; arithmetic dynamics; totally p-adic numbers}, language = {eng}, number = {3}, pages = {277-291}, title = {Heights and totally p-adic numbers}, url = {http://eudml.org/doc/286299}, volume = {171}, year = {2015}, }
TY - JOUR AU - Lukas Pottmeyer TI - Heights and totally p-adic numbers JO - Acta Arithmetica PY - 2015 VL - 171 IS - 3 SP - 277 EP - 291 AB - We study the behavior of canonical height functions $ĥ_{f}$, 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 $ĥ_{f}$ 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. LA - eng KW - height bounds; arithmetic dynamics; totally p-adic numbers UR - http://eudml.org/doc/286299 ER -