Wildly ramified power series with large multiplicity (Q2030220)

Let \(K\) be a field. The set \(\mathcal{A}(K)=\{f\in K[[z]]:f(0)\not=0\}\) with the operation of substitution forms a group. The subset \(\mathcal{N}(K)=\{f(z)\in\mathcal{A}(K):f'(0)=1\}\) is a normal subgroup of \(\mathcal{A}(K)\). In the case where char\((K)=p\) we say that \(\mathcal{N}(K)\) is the Nottingham group of \(K\). Let \(f(z)\in\mathcal{N}(K)\) with \(f(z)\not=z\). The residue fixed point index of \(f(z)\) is defined to be the residue of \(1/(z-f(z))\) at \(z=0\), and is denoted by index\((f)\). Thus index\((f)\) is the coefficient of \(z^{-1}\) in the expansion of \(1/(z-f(z))\) as a formal Laurent series. It is easily seen that index\((f)\) is invariant under conjugation by elements of \(\mathcal{A}(K)\). This paper investigates some generalizations of the residue fixed point which are defined when char\((K)=p\). Assume that char\((K)=p\) and let \(f(z)\in\mathcal{N}(K)\). Let \(q=v_z(f(z)-z)-1\) be the depth of \(f(z)\), let \(\ell\) be the least nonnegative residue of \(q\) modulo \(p\), and set \(r=(q-\ell)/p\). For \(1\le j\le r+1\) define the \(j\)th residue fixed point \(\tau_j(f)\) of \(f(z)\) to be the residue at \(z=0\) of \(z^{(r+1-j)p}/(z-f(z))\). Thus index\((f)=\tau_{r+1}(f)\). It is proved here that if \(1\le j\le r+1\) is minimum such that \(\tau_j(f)\not=0\) then \(\tau_j(h\circ f\circ h^{-1})=\tau_j(f)\) for all \(h\in\mathcal{N}(K)\). Assume now that \(p\nmid q\), \(q>p\), and \(1\le j\le r\). The author proves that \(j\) is minimum with \(\tau_j(f)\not=0\) if and only if the depth of \(f^{p^n}\) is equal to \((\ell+(j-1)p)(1+p+\cdots+p^{n-1})+qp^n\) for all \(n\ge1\).