# Newton Polygons

Since tropical curves are purely combinatorial objects, their systematic use can often reduce problems in algebraic geometry to combinatorics. In order to then answer these combinatorial questions, it is sometimes helpful to take a different point of view: that of Newton polygons and their polyhedral decompositions. In a sense this picture is dual to the description of tropical curves as rational graphs.

We begin with a definition. If ${\displaystyle f\in k[x,y]}$ is a polynomial (over any field or indeed any ring) we define the Newton polygon of ${\displaystyle f}$ to be the convex hull of those multi-indices whose corresponding cofficients are nonzero in ${\displaystyle f}$

${\displaystyle {\text{Newt}}(f)={\text{conv}}\left\{(i,j)\in \mathbb {N} ^{2}:a_{ij}\neq 0\right\}}$
(where ${\displaystyle a_{ij}}$ is the coefficient of ${\displaystyle x^{i}y^{j}}$ in ${\displaystyle f}$). This is a lattice polygon in ${\displaystyle \mathbb {N} ^{2}}$. If ${\displaystyle f}$ is generic of degree ${\displaystyle d}$, then ${\displaystyle {\text{Newt}}(f)}$ will be a right triangle of side length ${\displaystyle d}$.

If ${\displaystyle P\in \mathbb {T} }$ is a tropical polynomial, we define a polyhedral decomposition of ${\displaystyle {\text{Newt}}(P)}$ as follows. Each lattice point of ${\displaystyle {\text{Newt}}(P)}$ corresponds to a input in ${\displaystyle P=\max\{-\}}$. We put an edge between two such lattice points if the corresponding inputs are equal at some point (that is, if the corresponding regions in ${\displaystyle \mathbb {R} ^{2}}$ intersect along a wall of the tropical curve). Similarly, we put in ${\displaystyle 2}$-dimensional faces bounding a set of edges if the corresponding walls in the tropical curve meet at a vertex.

This gives us a polyhedral decomposition of ${\displaystyle {\text{Newt}}(P)}$. It follows immediately from the construction that the dual polyhedral decomposition is just the tropical curve associated to ${\displaystyle P}$.

Example 14.6

Consider the generic line: the Newton polygon is a right triangle of side length ${\displaystyle 1}$, so of course the polyhedral decomposition is trivial. Dualising we obtain the tropical curve we have seen before:

Example 14.7

Consider the generic conic: the Newton polygon is a right triangle of side length ${\displaystyle 2}$, and the polyhedral decomposition is the maximally fine triangulation. Dualising we obtain the tropical curve:

On its own this construction is not terribly useful, since we require full knowledge of the tropical curve in order to determine the polyhedral decomposition of ${\displaystyle {\text{Newt}}(P)}$. Better would be if we could obtain some alternative description of the polyhedral decomposition, since then we could dualise to obtain the tropical curve, practically for free.

As it turns out there is such a description. We begin by labeling each lattice point of ${\displaystyle {\text{Newt}}(P)}$ by the coefficient of the corresponding monomial in ${\displaystyle P}$. These determine a convex piecewise-linear function on the polygon, by requiring that at each lattice point the value of the function equals the label.

This then determines a polyhedral decomposition of ${\displaystyle {\text{Newt}}(P)}$: the ${\displaystyle 2}$-dimensional faces of the decomposition are precisely the regions on which the function is linear. It can be shown that this decomposition is the same as the one defined previously.

Example 14.8

Consider again the case of a generic conic. We have the labeled Newton polygon

and if we choose our coefficients generic enough, the corresponding function will only be linear on the small subtriangles of the following decomposition:

Dualising this we recover the tropical curve of a generic conic as in Example 14.7.

Example 14.9

On the other hand, if we had chosen our coefficients non-generically (for instance if we set ${\displaystyle a_{00}=a_{01}=a_{10}=a_{11}}$) we could obtain the following:

This is obtained from the previous polyhedral decomposition by fusing together two of the sub-polygons; we think of this as some sort of degeneration of the generic case. The tropical curve we get is: