Residual polynomials (reviewed)

Let $L/K$ be a totally ramified extension of $p$-adic fields, let $\alpha$ be a uniformizer of $L$ with minimal polynomial $\varphi$ over $K$ and denote by $v_\alpha$ the (exponential) valuation that is normalized such that $v_\alpha(\alpha)=1$. Let $\underline{L}$ be the residue field of $L$, and for $\delta \in \mathcal{O}_L$ write $\underline{\delta}$ for the image of $\delta$ in $\underline{L}$. Let $$\rho(x)=\sum_i \rho_i x^i\in\mathcal{O}_L[x]$$ be the ramification polynomial of $L/K$. Let $S$ be a segment of the ramification polygon of $L/K$ with length $l$, endpoints $(-k-l,v_\alpha(\rho_{k+l}))$, $(-k,v_\alpha(\rho_k))$, and slope $h/m=\left(v_\alpha(\rho_k)-v_\alpha(\rho_{k+l})\right)/l$. Then $$ A(x)=\sum_{j=0}^{l/m}\underline{\rho_{jm+k}\alpha^{jh-v_\alpha(\rho_k)}}x^{j}\in\underline{L}[x] $$ is called the residual polynomial associated to $\rho$ and $S$. If $S$ has positive slope then it follows from this construction that $x^kA(x^m)$ is an additive polynomial.

Residual polynomials (originally called associated polynomials) were introduced by Ore in 1928 [10.1007/BF01459087]. The residual polynomials of $\varphi$ are the residual polynomials associated to the segments $S_1, \dots, S_\ell$ of the ramification polygon of $\varphi$. Although the list of residual polynomials associated to $\varphi$ is not an invariant of $L/K$, the definition can be modified to define an invariant.

For $1\le i\le \ell$, let $h_i/m_i$ be the slope of $S_i$ and $A_i(x)$ its residual polynomial. Set $$ \mathcal{A}= \left\{ \left(\gamma_{\delta,1}{A_1}(\underline\delta^{h_1} x),\dots, \gamma_{\delta,\ell}{A_\ell}(\underline\delta^{h_\ell} x)\right) : \underline\delta\in\underline{K}^\times \right\},$$ where $\gamma_{\delta,\ell}=\underline\delta^{-h_\ell\deg A_\ell},$ and $ \gamma_{\delta,i}=\gamma_{\delta,i+1}\underline\delta^{-h_i\deg A_i}$ for $1\le i\le \ell-1$. Then $\mathcal{A}$ is an invariant of the extension $K[x]/(\varphi)$. We say that $\mathcal{A}$ is the equivalence class of residual polynomials associated to $L/K$.