Herbrand function of a family (reviewed)

Let $I/K$ be a family of extensions of a $p$-adic field $K$. We can express $I$ in terms of rams as $I=(r_1,\dots,r_w)_{\epsilon}^f$, in terms of Swan slopes as $I=[s_1,\dots,s_w]_{\epsilon}^f$ or in terms of the means $\langle m_1,\dots,m_w\rangle_\epsilon^f$. The Herbrand function of $I/K$ is the piecewise linear function $\phi_{I/K}:\R_{\ge0}\rightarrow\R_{\ge0}$ whose graph is obtained by joining the points $(0,0),(r_1,s_1),\dots,(r_w,s_w)$ with line segments; in addition, there is a ray with slope $1/(\epsilon p^w)$ starting at $(r_w,s_w)$. It follows that for $1\le i\le w$ the segment whose right endpoint is $(r_i,s_i)$ has slope $1/(\epsilon p^{i-1})$.

The rams of $I/K$ are labeled in blue on the horizontal axis of the diagram. The wild Swan slopes of $I/K$ are labeled in black on the vertical axis of the diagram; to the left of each Swan slope is a rescaling by a factor $\epsilon p^w$. The green numbers along the vertical axis are the means. The rescaled green numbers on the left which correspond to intersections of the vertical axis with extensions of segments of the graph of $\phi_{I/K}$ give some of the indices of inseparability of the extensions $L/K$ in $I/K$.