A Note on tropical curves and the Newton diagrams of plane curve singularities

Tropical geometry is rapidly developing as a new study area in mathematics. In [7], Mikhalkin counts nodal curves on toric surfaces, which is an epoch-making result in algebraic geometry. Though there are several studies concerning tropical curves corresponding to singular algebraic curves, see for instance [10, 6, 2], the study of relation between theory of tropical curves and the singularity theory is still underdeveloping in tropical geometry.

In singularity theory, the Newton diagram is an important tool to get information of singularities. Let $f$ be a polynomial of two variables over $C$, and suppose that $f(0,0)=0$ and $f$ has an isolated singularity at $0=(0,0)\in C^2$. Set $f(x,y)={\sum }_{(i,j)}{a}_{ij}{x}^i{y}^j$, and ${\Delta }_f=Conv\left(\right\{(i,j)\in R^2;a_{ij}\ne 0\left\}\right)$, where $Conv(\cdot )$ is the convex hull. The convex hull ${\Delta }_f$ is called the Newton polytope of $f$. Let ${\Gamma }_{-}\left(f\right)$ be the polyhedron defined by

where $Closure(\cdot )$ is the closure with usual topology of $R^2$. The Newton boundary $\Gamma \left(f\right)$ of $f$ is the union of compact faces of $Conv\left(\right\{(i,j)+(R_{\ge 0}{)}^2;a_{ij}\ne 0\})$. The singularity $(f,0)$ is convenient if ${\Gamma }_{-}\left(f\right)$ is compact. The singularity $(f,0)$ is Newton non-degenerate if, for any face $\sigma$ in $\Gamma \left(f\right)$, the function $f^{\sigma }(x,y):={\sum }_{(i,j)\in \sigma \cap {\Delta }_f}{a}_{ij}{x}^i{y}^j$ has no singularity in $(C^{\ast }{)}^2$.

Note that, for a convenient and Newton non-degenerate singularity, the Milnor number of $(f,0)$ is computed from ${\Gamma }_{-}\left(f\right)$, which is the celebrated theorem of Kouchnirenko [5]. In this paper, we regard ${\Gamma }_{-}\left(f\right)$ as a part of polyhedrons obtained as a dual subdivision of a tropical curve and give a meaning of ${\Gamma }_{-}\left(f\right)$ from the viewpoint of tropical geometry.

To state our result, we here introduce some terminologies in tropical geometry. Let $F$ be a polynomial of two variables over the field $K$ of convergent Puiseux series over $C$. The tropical curve $T_F$ is defined by the image of $F=0$ by the valuation map, which is a 1-simplicial complex in $R^2$. The valuation of the coefficients of $F$ induces a subdivision $Sd\left(F\right)$ of the Newton polytope ${\Delta }_F$ and it is known that $Sd\left(F\right)$ is dual to $T_F$, which is so-called the Duality Theorem (see §2).

Now we consider a union of polygons corresponding to a part of polygons of $Sd\left(F\right)$. Let ${\Delta }^{\prime }$ be a sub-polyhedron of ${\Delta }_F$. A subset $S$ of $T_F$ is called the tropical sub-curve with respect to ${\Delta }^{\prime }$ if ${\Delta }^{\prime }$ is a union of sub-polyhedrons of $Sd\left(F\right)$ and $S$ is dual to the subdivision of ${\Delta }^{\prime }$ induced by $Sd\left(F\right)$. We denote it by $T_F{|}_{{\Delta }^{\prime }}$. Note that if ${\Delta }^{\prime }={\Delta }_F$ then $T_F{|}_{{\Delta }^{\prime }}=T_F$. See Definition 3.1 for the precise definition of $T_F{|}_{{\Delta }^{\prime }}$.

For plane curve singularities, the real morsification due to A’Campo [1] and Gusein-Zade [3] gives an explicit way to understand mutual positions of vanishing cycles. Our hope is that we can perform the same observation for tropical curves realized in ${\Gamma }_{-}\left(f\right)$. The main theorem in this paper asserts that we can see the vanishing cycles of $f=0$ on the tropical curve in ${\Gamma }_{-}\left(f\right)$. For a tropical sub-curve $T_F{|}_{{\Delta }^{\prime }}$, let $v\left(T_F{|}_{{\Delta }^{\prime }}\right)$ denote the number of 4-valent vertices of $T_F{|}_{{\Delta }^{\prime }}$ and $r\left(T_F{|}_{{\Delta }^{\prime }}\right)$ denote the number of regions bounded by $T_F{|}_{{\Delta }^{\prime }}\subset R^2$.

This result is an analogy of the following equality required for a real morsification:

where $f_s$ is a real morsification of $f$, $\delta \left(f_s\right)$ is the number of double points of $f_s=0$ in a previously fixed small neighborhood $U$ of the origin, and $r\left(f_s\right)$ is the number of bounded regions of $\{f_s=0\}{|}_{R^2}\cap U$. As a corollary of Theorem 1.1 we have the equality $\delta \left(f\right)=v\left(T_F{|}_{{\Gamma }_{-}\left(f\right)}\right)$, where $\delta \left(f\right)$ is the number of double points of $(f,0)$, see Corollary 3.5.

Remark that the polynomial $F$ in Theorem 1.1 is given by a patchworking polynomial asssociated with a subdivision of ${\Delta }_F$. A similar observation appears in a paper of Shustin [9], where a real polynomial whose critical points has a given index distribution is constructed.

We organize the paper as follows. In section 2, we introduce tropical curves and subdivisions of Newton polytopes induced from the valuation map and state the Duality Theorem. In section 3, we give the definition of tropical sub-curves and prove Theorem 1.1. Two examples will be given before the proof.

First, we give some definitions about polytopes. A polygon in $R^2$ is the intersection of a finite number of half-spaces in $R^2$ whose vertices are contained in the lattice $Z^2\subset R^2$. A polygon is called a polytope if it is compact. In this paper, a polyhedron means a union of polytopes which is connected and compact. Thus a polyhedron is not convex generally. A subset in a polyhedron is a sub-polyhedron if it is a polyhedron as a subset of $R^2$. In particular, if a sub-polyhedron is a polytope then the sub-polyhedron is called a sub-polytope.

Let $K:=C\left\{\right\{t\left\}\right\}$ be the field of convergent Puiseux series over $C$, and denote the usual non-trivial valuation on $K$ by $val:K^{\ast }\to R$, that is,

where $K^{\ast }:=K\setminus \left\{0\right\}$ and $b_{k_0}\ne 0$.

For a reduced polynomial

over $K$, we denote by $Supp\left(F\right)$ and ${\Delta }_F$ the support of $F$ and the Newton polytope of $F$, respectively, that is, $Supp\left(F\right):=\left\{\right(i,j)\in R^2;c_{ij}\ne 0\}$ and ${\Delta }_F=Conv\left(Supp\right(F\left)\right)\subset R^2$. Throughout this paper, we assume that the Newton polytope ${\Delta }_F$ of $F$ is 2-dimensional.

The valuation map $Val:(K^{\ast }{)}^2\to R^2$ is defined by $(z,w)\mapsto \left(val\right(z),val(w\left)\right)$, which is a homomorphism.

Remark that a tropical curve $T_F$ has a structure of a $1$-dimensional simplicial complex, cf.[7]. We call a $1$-simplex an edge and a $0$-simplex a vertex as usual.

Let $F$ be a polynomial over $K$. We introduce the Duality Theorem which gives a correspondence between the tropical curve $T_F$ defined by $F$ and the Newton polytope ${\Delta }_F$ of $F$. Let ${\Delta }_{\nu }\left(F\right)$ be the 3-dimensional polygon defined by

and ${\nu }_F:{\Delta }_F\to R$ be the function defined by

which is a continuous piecewise linear convex function. We can get the following three kinds of sub-polytopes of ${\Delta }_F$ from ${\nu }_F$:

• linearity domains of ${\nu }_F$: ${\Delta }_1,\cdots ,{\Delta }_N$,

• 1-dimensional polytopes: ${\Delta }_i\cap {\Delta }_j\ne \varnothing$ and $\ne \left\{pt\right\}$,

• 0-dimensional polytopes: ${\Delta }_{i_1}\cap {\Delta }_{i_2}\cap {\Delta }_{i_3}\ne \varnothing$,

where a linerity domain of ${\nu }_F$ means a maximal sub-polytope $R$ of the domain ${\Delta }_F$ such that the restriction ${\nu }_F{|}_R$ is an affine linear function. These polytopes give a subdivision of ${\Delta }_F$, which we denote by $Sd\left(F\right)$. In particular, we call a 1-dimensional and a 0-dimensional polytope a vertex and a edge of $Sd\left(F\right)$, respectively.

For edges of a tropical curve, a certain weight $w:\left(\text{edges of}T\right)\to N$ is defined by directional vectors of edges canonically. We omit the definition since we don’t use it in this paper.

We can find the following claim in §2.5.1 of [4]. A proof in general dimension can be found in [7].

We call the subdivision $Sd\left(F\right)$ the dual subdivision of $T_F$. By Theorem 2.2, we can regard a tropical curve as a dual subdivision of the Newton polytope of its defining polynomial.

In this section, we first introduce the precise definition of tropical sub-curves which we mentioned in the introduction. Let $F\in K[x,y]$ be a polynomial over $K$ and

be the dual subdivision of $T_F$. Because of the structure theorem in tropical geometry, a tropical hypersurface has the structure of a polyhedral complex. In particular, a plane tropical curve is an embeded plane graph in $R^2$.

Let $[u,v]$ denote the edge of the tropical curve $T_F$ whose endpoints are 0-cells $u$ and $v$. Set $[u,v)=[u,v]\setminus \left\{v\right\}$. We allow that one of the endpoints is at $\infty$. In this case, the other endpoint is contained in the 0-cells of the curve. Note that $[u,\infty ]=[u,\infty )$.

Let

be a subset of $Sd\left(F\right)$ such that $\bigcup {Sd}^{\prime }\left(F\right)\setminus {Sd}^{\left[0\right]}\left(F\right)$ is connected, where ${Sd}^{\left[0\right]}\left(F\right)$ is the set of vertices of $Sd\left(F\right)$. Let ${\Delta }^{\prime }$ be a sub-polyhedron of ${\Delta }_F$ given as the union of ${Sd}^{\prime }\left(F\right)$.

Let $V=\{v_1,\dots ,v_N\}$ and $E=\left\{\right[u,v];u,v\in V\}$ be the set of vertices and edges of $T_F$ respectively.

Suppose that $f$ is convenient. For the lattice points $(i,j)\in {\Gamma }_{-}\left(f\right)\cap Z^2$, we define a map ${\nu }_f{|}_{{\Gamma }_{-}\left(f\right)\cap Z^2}:{\Gamma }_{-}\left(f\right)\cap Z^2\to R$ by

where $\{a_k{\}}_{k\in N},\{b_k{\}}_{k\in N}$ are non-negative strictly increasing sequences of integers. We then extend it to the whole domain ${\Gamma }_{-}\left(f\right)$ as a continuous piecewise linear function and obtain a map ${\nu }_f:{\Gamma }_{-}\left(f\right)\to R$. Taking sufficiently large values for ${\nu }_f$ at the lattice points of the Newton boundary of $f$, we may assume that the other sub-polytopes are triangles with area $1/2$.