Let K K be a complete discrete valuation field with residue class field k k , where both are of positive characteristic p p . Then the group of wild automorphisms of K K can be identified with the group under composition of formal power series over k k with no constant term and X X -coefficient 1 1 . Under the hypothesis that p > b 2 p > b^2 , we compute the first nontrivial coefficient of the p p th iterate of a power series over k k of the form f = X + ∑ i ≥ 1 a i X b + i f = X + \sum _{i \geq 1} a_iX^{b+i} . As a result, we obtain a necessary and sufficient condition for an automorphism to be “ b b -ramified”, having lower ramification numbers of the form i n ( f ) = b ( 1 + ⋯ + p n ) i_n(f) = b(1 + \cdots + p^n) . This is a vast generalization of Nordqvist’s 2017 theorem on 2 2 -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 C p \mathbf {C}_p 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 2 2 -ramified power series.

Let K be a complete discrete valuation field with residue class field k, where both are of positive characteristic p. Then the group of wild automorphisms of K can be identified with the group under composition of formal power series over k with no constant term and X-coefficient 1. Under the hypothesis that p > b 2 , we compute the first nontrivial coefficient of the pth iterate of a power series over k of the form f = X + i≥1 a i X b+i . As a result, we obtain a necessary and sufficient condition for an automorphism to be "b-ramified," having lower ramification numbers of the form in(fThis is a vast generalization of Nordqvist's 2017 theorem on 2-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 Cp 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 2-ramified power series.

Abstract. The best previous lower bounds for kissing numbers in dimensions 25-31 were constructed using a set S with |S| = 480 of minimal vectors of the Leech Lattice, Λ 24 , such that x, y ≤ 1 for any distinct x, y ∈ S. Then, a probabilistic argument based on applying automorphisms of Λ 24 gives more disjoint sets S i of minimal vectors of Λ 24 with the same property. Cohn, Jiao, Kumar, and Torquato proved that these subsets give kissing configurations in dimensions 25-31 of given size linear in the sizes of the subsets. We achieve |S| = 488 by applying simulated annealing. We also improve the aforementioned probabilistic argument in the general case. Finally, we greedily construct even larger S i 's given our S of size 488, giving increased lower bounds on kissing numbers in R 25 through R 31 .