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Ramification of wild automorphisms of Laurent series fields
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15250 Kenz Kallal , Hudson Kirkpatrick
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15250 Kenz Kallal , Hudson Kirkpatrick
Abstract:Let 
be a complete discrete valuation field with residue class field 
, where both are of positive characteristic 
. Then the group of wild automorphisms of 
can be identified with the group under composition of formal power series over 
with no constant term and 
-coefficient 
. Under the hypothesis that 
, we compute the first nontrivial coefficient of the 
th iterate of a power series over 
of the form 
. As a result, we obtain a necessary and sufficient condition for an automorphism to be ``
-ramified'', having lower ramification numbers of the form 
. This is a vast generalization of Nordqvist's 2017 theorem on 
-ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in 
and a 2015 result of Lindahl-Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl-Nordqvist's 2018 theorem bounding the norm of periodic points of 
-ramified power series.
中文翻译：
Laurent系列场的野生自同构的分支
摘要：让具有残值类别字段的完全离散估值字段都具有正特征。然后，可以将的野生自同构组与形式幂级数组成下的组（没有常数项和系数）一起识别。在以下假设下，我们计算形式的幂级数的迭代的第一个非平凡系数。结果，我们获得了具有同等形式的较低分支数的自同构``分支''的必要和充分条件。这是Nordqvist 2017年定理关于 分叉的幂级数，以及最小分叉的幂级数的类似结果，这些结果对于Lindahl于2013年在线性化圆盘上的论文和Lindahl-Rivera-Letelier于2015年在非档案领域的最佳循环中的算术动力学很有用残留特征为正。我们计算的成功也有望推动Lindahl-Nordqvist 2018年定理的推广，该定理以-幂次幂的周期点的范数为界。
更新日期：2021-02-08
be a complete discrete valuation field with residue class field , where both are of positive characteristic . Then the group of wild automorphisms of can be identified with the group under composition of formal power series over with no constant term and -coefficient . Under the hypothesis that , we compute the first nontrivial coefficient of the th iterate of a power series over of the form . As a result, we obtain a necessary and sufficient condition for an automorphism to be ``-ramified'', having lower ramification numbers of the form . This is a vast generalization of Nordqvist's 2017 theorem on -ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in and a 2015 result of Lindahl-Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl-Nordqvist's 2018 theorem bounding the norm of periodic points of -ramified power series. 中文翻译：
Laurent系列场的野生自同构的分支
摘要：让
具有残值类别字段的完全离散估值字段都具有正特征。然后，可以将的野生自同构组与形式幂级数组成下的组（没有常数项和系数）一起识别。在以下假设下，我们计算形式的幂级数的迭代的第一个非平凡系数。结果，我们获得了具有同等形式的较低分支数的自同构``分支''的必要和充分条件。这是Nordqvist 2017年定理关于 分叉的幂级数，以及最小分叉的幂级数的类似结果，这些结果对于Lindahl于2013年在线性化圆盘上的论文和Lindahl-Rivera-Letelier于2015年在非档案领域的最佳循环中的算术动力学很有用残留特征为正。我们计算的成功也有望推动Lindahl-Nordqvist 2018年定理的推广，该定理以-幂次幂的周期点的范数为界。 
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