Ramification polygon diagram (reviewed)

The blue graph shows the ramification polygon.

• The blue number to the right of each segment is its slope.
• Coordinates are given for the black points at $(-p^j, i_j)$ where $[i_0,...,i_\nu]$ are the indices of inseparability. These points include the vertices of the ramification polygon. In some cases there are additional points on or above the polygon.
• Each residual polynomial $A(z)$ of $L/K$ is associated to a segment $S$ of the ramification polygon; if $S$ has positive slope then the coefficients of the polynomial come from the black points which lie on the segment. More precisely, assume that $S$ has right endpoint $(-p^k,i_k)$, left endpoint $(-p^l,i_l)$, and slope $h/m>0$. Let $k\le j\le l$. If there is a black point on $S$ with first coordinate $-p^j$ then the coefficient of $x^{p^j}$ in the additive polynomial $x^{p^k}A(x^m)$ is given by the label attached to this point. Otherwise the coefficient of $x^{p^j}$ is 0.
• Since successive segments of the ramification polygon share a vertex, the constant term of each residual polynomial is equal to the leading coefficient of the next.