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, and provide a closed formula for it. 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.

在本文中，我们考虑了分叉的幂级数，即在正特性字段上定义的幂级数，它固定了原点，与原点相切。在这种情况下，我们引入了一个在坐标变化下的新不变量，称为第二个残基不动点索引，并为其提供了一个封闭式。顾名思义，该不变量与残基不动点指数密切相关，并且在幂级数具有较小多重性的情况下它们是一致的。最后，根据这个新的不变性，我们描述了具有大多重性的幂级数，该幂级数在迭代下的原点处具有最小的多重性。