Residual polynomials (awaiting review)

Let $L/K$ be a totally ramified extension of a $p$-adic field $K$, 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 $v_\alpha(\alpha)=1$. Set $\underline{L}$ to be the residue class field of $L$, and for $\delta \in \mathcal{O}_L$ write $\underline{\delta}$ for the image of $\delta$ in $\underline{L}$. For any polynomial $$\rho(x)=\sum_i \rho_i x^i\in\mathcal{O}_L[x]$$ let $S$ be a segment of the Newton polygon of $\rho$ of length $l$ with end points $(k,v_\alpha(\rho))$ and $(k+l,v_\alpha(\rho_{k+l}))$, and slope $-h/e=\left(v_\alpha(\rho_{k+l})-v_\alpha(\rho_k)\right)/l$. Then $$ A(x)=\sum_{j=0}^{l/e}\underline{\rho_{je+k}\alpha^{jh-v_\alpha(\rho_k)}}x^{j}\in\underline{L}[x] $$ is called the residual polynomial associated to $\rho$ and $S$.

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 $-h_i/e_i$ be the slope of $S_i$ and $A_i(x)$ its residual polynomial. Then $$ \mathcal{A}= \left\{ \left(\gamma_{\delta,1}{A_1}(\underline\delta^{h_1} x),\dots, \gamma_{\delta,\ell}{A_\ell}(\underline\delta^{h_\ell} x)\right) : \underline\delta\in\underline{K}^\times \right\}$$ 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 $1\le i\le \ell-1$ is an invariant of the extension $K[x]/(\varphi)$. We call $\mathcal{A}$ the residual polynomial classes of $L$.