Newton polygon of a polynomial (reviewed)

Let $K$ be a nonarchimedean local field with discrete valuation $v_K$, and $f \in K[x]$, given by $$ f(x) = a_0 + a_1 x + \cdots + a_nx^n,$$ and such that $a_0a_n \neq 0$.

Then the Newton polygon of $f$ is the lower convex hull of the set of points $(i,v_K(a_i))$, if we ignore all points with $a_i=0$.

In the context of isogeny classes of abelian varieties over finite fields, the constant term $a_0=1$ always, and therefore the Newton polygon begins at $(0,0)$. In this context, the LMFDB accepts two methods to enter Newton polygons:

• The first is to give a list of slopes $[s_1,s_2,s_3, \ldots]$. This denotes the Newton polygon that, starting at $(0,0)$, is given by a line segment of width 1 with slope $s_1$, followed by a line segment of width 1 with slope $s_2$, etc. Note that it is possible for a slope to appear multiple times if the Newton polygon has the same slope for a width of more than 1. Note also that because the Newton polygon is a convex hull, the slopes $s_i$ will necessarily be increasing.

• The second is to give a list of points, called "breaks," $[(x_1,y_1),(x_2,y_2), (x_3,y_3), \ldots]$, for $0 < x_1 < x_2 < x_3 < \ldots$. This denotes the Newton polygon that, starting at $(0,0)$, has constant slope from $(0,0)$ to $(x_1,y_1)$, then has constant (but different) slope from $(x_1,y_1)$ to $(x_2,y_2)$, etc. Note that by the definition of the Newton polygon, each $x_i$ is a positive integer.