Wildly ramified power series with large multiplicity

Abstract

In this paper we consider wildly ramified power series, i.e., power series defined over a field of positive characteristic, fixing the origin, where it is tangent to the identity. In this setting we introduce a new invariant under change of coordinates called the second residue fixed point index. As the name suggests this invariant is closely related to the residue fixed point index, and they coincide in the case that the power series have small multiplicity. Finally, we characterize power series with large multiplicity having the smallest possible multiplicity at the origin under iteration, in terms of this new invariant.

Residue fixed point index and wildly ramified power series

“…A direct computation using (1.6) shows that résit(f ) = q + 1 2 + (−1) q b q a q+1 = 0, v The situation is now clear form the recent characterization of (p + 1)-ramification by the first named author in [Nor19].…”