A Note on tropical curves and the Newton diagrams of plane curve singularities
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko’s formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A’Campo and Gusein-Zade.
Tropical geometry is rapidly developing as a new study area in mathematics. In , 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 be a polynomial of two variables over , and suppose that and has an isolated singularity at . Set , and , where is the convex hull. The convex hull is called the Newton polytope of . Let be the polyhedron defined by
where is the closure with usual topology of . The Newton boundary of is the union of compact faces of . The singularity is convenient if is compact. The singularity is Newton non-degenerate if, for any face in , the function has no singularity in .
Note that, for a convenient and Newton non-degenerate singularity, the Milnor number of is computed from , which is the celebrated theorem of Kouchnirenko . In this paper, we regard as a part of polyhedrons obtained as a dual subdivision of a tropical curve and give a meaning of from the viewpoint of tropical geometry.
To state our result, we here introduce some terminologies in tropical geometry. Let be a polynomial of two variables over the field of convergent Puiseux series over . The tropical curve is defined by the image of by the valuation map, which is a 1-simplicial complex in . The valuation of the coefficients of induces a subdivision of the Newton polytope and it is known that is dual to , which is so-called the Duality Theorem (see §2).
Now we consider a union of polygons corresponding to a part of polygons of . Let be a sub-polyhedron of . A subset of is called the tropical sub-curve with respect to if is a union of sub-polyhedrons of and is dual to the subdivision of induced by . We denote it by . Note that if then . See Definition 3.1 for the precise definition of .
For plane curve singularities, the real morsification due to A’Campo  and Gusein-Zade  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 . The main theorem in this paper asserts that we can see the vanishing cycles of on the tropical curve in . For a tropical sub-curve , let denote the number of 4-valent vertices of and denote the number of regions bounded by .
For any Newton non-degenerate and convenient isolated singularity , there is a polynomial such that and satisfies
This result is an analogy of the following equality required for a real morsification:
where is a real morsification of , is the number of double points of in a previously fixed small neighborhood of the origin, and is the number of bounded regions of . As a corollary of Theorem 1.1 we have the equality , where is the number of double points of , see Corollary 3.5.
Remark that the polynomial in Theorem 1.1 is given by a patchworking polynomial asssociated with a subdivision of . A similar observation appears in a paper of Shustin , 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.
I am grateful to Masaharu Ishikawa and Takeo Nishinou for fruitful discussions. I would like to thank Eugenii Shustin for precious comments and telling me about his previous result. I also thank Nikita Kalinin for giving me interesting comments.
First, we give some definitions about polytopes. A polygon in is the intersection of a finite number of half-spaces in whose vertices are contained in the lattice . 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 . In particular, if a sub-polyhedron is a polytope then the sub-polyhedron is called a sub-polytope.
Let be the field of convergent Puiseux series over , and denote the usual non-trivial valuation on by , that is,
where and .
For a reduced polynomial
over , we denote by and the support of and the Newton polytope of , respectively, that is, and . Throughout this paper, we assume that the Newton polytope of is 2-dimensional.
The valuation map is defined by , which is a homomorphism.
with usual topology on is called the (plane) tropical curve defined by .
Remark that a tropical curve has a structure of a -dimensional simplicial complex, cf.. We call a -simplex an edge and a -simplex a vertex as usual.
Let be a polynomial over . We introduce the Duality Theorem which gives a correspondence between the tropical curve defined by and the Newton polytope of . Let be the 3-dimensional polygon defined by
and be the function defined by
which is a continuous piecewise linear convex function. We can get the following three kinds of sub-polytopes of from :
linearity domains of : ,
1-dimensional polytopes: and ,
0-dimensional polytopes: ,
where a linerity domain of means a maximal sub-polytope of the domain such that the restriction is an affine linear function. These polytopes give a subdivision of , which we denote by . In particular, we call a 1-dimensional and a 0-dimensional polytope a vertex and a edge of , respectively.
For edges of a tropical curve, a certain weight is defined by directional vectors of edges canonically. We omit the definition since we don’t use it in this paper.
Theorem 2.2 (Duality Theorem).
is dual to the tropical curve
in the following sense:
(1) the components of are in 1-to-1 correspondence with the vertices of the subdivision ,
(2) the edges of are in 1-to-1 correspondence with the edges of the subdivision so that an edge is dual to an orthogonal edge of the subdivision , having the lattice length equal to , which is a weight of ,
(3) the vertices of are in 1-to-1 correspondence with the polytopes
so that the valency of a vertex of is equal to the number of sides of the dual polygon.
We call the subdivision the dual subdivision of . By Theorem 2.2, we can regard a tropical curve as a dual subdivision of the Newton polytope of its defining polynomial.
3. Main Results
In this section, we first introduce the precise definition of tropical sub-curves which we mentioned in the introduction. Let be a polynomial over and
be the dual subdivision of . 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 .
Let denote the edge of the tropical curve whose endpoints are 0-cells and . Set . We allow that one of the endpoints is at . In this case, the other endpoint is contained in the 0-cells of the curve. Note that .
be a subset of such that is connected, where is the set of vertices of . Let be a sub-polyhedron of given as the union of .
Let and be the set of vertices and edges of respectively.
A subset of
is called the
with respect to
if it has the structure of
the metric (open) sub-graph
of the tropical curve
which satisfies the following conditions:
(1) the set of vertices is given by ,
(2) the set of edges is given by the following manners: for each ,
if then ,
if and then ,
if and then , where is taken as the middle point of .
We denote the tropical sub-curve of with respect to by .
(1) Let be a polynomial over given by
The Newton polytope of is . See on the left in Figure 1. This polynomial is in Theorem 1.1 for the singularity of at the origin. The polyhedron in the figure is for the singularity . The tropical sub-curve with respect to is as shown on the right. Since and , the equality in Theorem 1.1 is verified.
(2) Let be a polynomial over given by
The Newton polytope of is See on the left in Figure 2. This polynomial is in Theorem 1.1 for the singularity of at the origin. The polyhedron in the figure is for the singularity . The tropical sub-curve with respect to is as shown on the right. Since and , the equality in Theorem 1.1 holds.
Suppose that is convenient. For the lattice points , we define a map by
where are non-negative strictly increasing sequences of integers. We then extend it to the whole domain as a continuous piecewise linear function and obtain a map . Taking sufficiently large values for at the lattice points of the Newton boundary of , we may assume that the other sub-polytopes are triangles with area .
We call the subdivision of defined as above the special subdivision of and each square in this subdivision the special square.
Let be coprime integers. The number of special squares in the special subdivision of is .
Let be the rectangle given by
We consider the special subdivision of . We decompose it into vertical rectangles
The special subdivision of induces a special subdivision of each . Let be the segment connecting and . We denote by the number of special squares in which intersect . Similarly, we denote by the number of special squares in which intersect . Obviously .
Let be the number of special squares of . Notice that since . Thus it is enough to show . Without loss of generality, we may assume . Let , be integers such that and . The segment can be denoted as
Set . Then, is calculated as
where means the largest integer not greater than . Thus, we obtain
Proof of Theorem 1.1.
Choose a polynomial such that the Newton polytope coincides with . To decide coefficients of , we take the convex function as a linear extension of used in the definition of the special subdivision of , and define as the patchworking polynomial defined by , that is,
In the rest of the proof, we check satisfies the equality in the assertion. To calculate the number of special squares, we decompose into two sub-polyhedrons as follows. Let be coprime integers. We denote the intersection points of the Newton boundary of and the lattice by
where . We set and define
Notice that and . For , we define the subset of as
Then, decomposes as . For , we denote by the number of special squares contained in the special subdivision of induced by that of . Then, using Lemma 3.4, we have
Next we will show the following equalities:
By Theorem 2.2, the correspondence between subdivisions and tropical curves, introduced in Theorem 2.2, gives a 1-to-1 correspondence of parallerograms and 4-valent vertices. In our case, any 4-valent vertex corresponds to a special square. Thus, the number of special squares, , coincides with the number of 4-valent vertices of . Hence equality (1) holds.
We prove the other equality. There is a 1-to-1 correspondence between
where is the set of vertices of a special square in . Moreover, by the Duality Theorem 2.2 of tropical curves, we have a 1-to-1 correspondence between the bounded regions contained in the complement of and the interior lattice points in . Since
Thus equality (2) holds.
Set . From equality (1), we get as
For the Milnor number , we use Kouchnirenko’s formula in :
Thus, we get
Let be a polynomial obtained in Theorem 1.1. Then the number of double points of coincides with .
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