Refined count for rational tropical curves in arbitrary dimension

In this paper we introduce a refined multiplicity for rational tropical curves in arbitrary dimension, which generalizes the refined multiplicity introduced by F. Block and L. G\"ottsche in arXiv:1407.2901 . We then prove an invariance statement for the count of rational tropical curves in several enumerative problems using this new refined multiplicity. This leads to the definition of Block-G\"ottsche polynomials in any dimension.


Enumerative geometry, classical and tropical
Tropical enumerative geometry takes its roots in the founding paper [14] by G. Mikhalkin. He there proved a correspondence theorem that allows one to compute several classical geometric invariants for toric surfaces, namely Gromov-Witten type invariants and Welschinger invariants whenever defined, through the solving of some associated enumerative problem in planar tropical geometry. Other approaches to the correspondence theorem were later provided by E. Shustin [19] and I. Tyomkin [21] along with a generalization for rational curves in higher dimensional toric varieties by T. Nishinou and B. Siebert [17]. Some progress has even been made in the non-toric case, for instance in the case of abelian surfaces by T. Nishinou [16].
In each situation, the spirit of the tropical approach is the same: the goal is to make a count of algebraic curves satisfying some constraints and which is known to be invariant when the constraints vary in some family, e.g. a count of curves passing through some points and which does not depend on the choice of the points as long as it is generic. One then degenerates the constraints to the so-called tropical limit, where the classical enumerative problem becomes a tropical enumerative problem. The desired curve count might then be recovered by counting the tropical curves solution to this new problem with a suitable multiplicities depending on the problem.
Example 1.1. Explicitly, in the case of rational curves in the projective plane CP 2 , one of these enumerative problems consists in finding the degree d rational curves passing through a chosen configuration of 3d − 1 points chosen in CP 2 . The result does not depend on the chosen points if their choice is generic, and is known as N d , a Gromov-Witten invariant of CP 2 .
Over the real numbers, J-Y. Welschinger showed in [23] that if the configuration of 3d − 1 points is chosen invariant by conjugation and the curves are counted with a suitable sign, the count only depends on d and the number s of pairs of complex conjugated points. It is denoted by W d,s . In particular W d,0 corresponds to the signed count of real rational curves passing through a generic configuration of 3d − 1 real points. ♦ We keep in mind the case of degree d rational curves in the pursue of this introduction. To the tropical limit, the enumerative problems leading to the definition of N d and W d,0 degenerate to the following tropical enumerative problem: finding degree d rational tropical curves passing through a chosen configuration of 3d − 1 points. For a definition of tropical curves, see section 2, but for the reading of the introduction, one only needs to know that they are some finite graphs inside an affine space R r , and that they have some infinite edges whose slopes form the degree of the curve. According to [14], to recover both invariants N d and W d,0 , it suffices to count tropical curves Γ passing through a generic choice of 3d − 1 points with different multiplicities m C Γ and m R Γ , defined as follows: there exists a natural vertex multiplicity m V , and one then sets Both multiplicities are a product over the vertices of the tropical curve. The count with m C Γ gives N d and the count with m R Γ gives W d,0 . In [3], F. Block and L. Göttsche introduced a refined multiplicity for tropical curves, which is also a product over the vertices of the tropical curve: This multiplicity is a Laurent polynomial in one variable q, which specializes to m C Γ and m R Γ when q goes respectively to 1 and −1. The count of tropical solutions using this refined multiplicity was proven to also lead to an invariant by I. Itenberg and G. Mikhalkin in [12]. The classical meaning of these tropically defined invariants remains quite mysterious. Conjecturally [10], they relate to the computation of the χ −y -genera of the relative Hilbert schemes of some linear systems of curves. Some attempts in this direction have been made by J. Nicaise, S. Payne and F. Schroeter [16].
Another lead to understand these invariants was provided by Mikhalkin in [15]. We still consider the degree d rational curves in the case of the projective plane, and a configuration P of 3d− 1 points which are this time chosen on the coordinate axis of the projective plane. Mikhalkin introduced a signed count for the real oriented curves passing through ±p for each p ∈ P. This count is refined according to the value of a so called quantum index, which is defined for each oriented curve of type I. He then proved that the result of the count only depends on the number of pairs of complex conjugated points on each axis. If the points are all chosen real, he relates the refined count to the tropical invariant obtained solving the analog tropical enumerative problem, and using Block-Göttsche multiplicities. The tropical analog is finding degree d rational tropical curves whose infinite edges are contained in some fixed lines. The result was extended by the author in [7] and [6] in case there are complex conjugated points.
Since, many other tropical enumerative problems have been shown to support a refinement, i.e. the existence of a non-trivial polynomial multiplicity such that the count of solution using this multiplicity does not depend on the chosen constraints. See for instance [2], [18] or [9]. However, in every of these situations, the new refined multiplicity of a tropical curve is a product over its vertices, just as is the complex multiplicity. In higher dimension, the complex multiplicity is not necessarily a product over the vertices. This a priori prevents anyone to find a simple multiplicity which is a product over the vertices and leads to an invariant. Finding a refined multiplicity which is not a product over the vertices seems even more complicated. Here, we show an in-between approach, providing a multiplicity which is up to sign a product over the vertices. The definition of this sign, which is at the center of the proof, carries the non-vertex information.

Moment problem and refined invariance
A natural generalization of the tropical enumerative problem considered by Mikhalkin in [15] is the following, which we call moment problem. Let N be a lattice of rank r and M its dual lattice. From now on, N R = N ⊗ Z R will be our workspace for tropical curves and every enumerative problem. We consider tropical curves of degree ∆, which is a multiset of vectors in N of total sum 0, and not containing the zero vector. Let ω be a 2-form on N , and for each n e ∈ ∆, fix a hyperplane H e with slope ker ι ne ω. We look for degree ∆ rational tropical curves such that the unbounded end directed by n e is contained in H e . As this is not enough to ensure having a finite number of solutions, we add the condition that a chosen unbounded end e 0 is contained in a fixed plane D ⊂ N R transverse to H e 0 . Remark 1.2. In the planar case, there is only one possible choice of non-zero 2-form ω up to multiplication by a scalar, and no need to choose a plane D. For each choice of ∆, the moment problems coincide with the enumerative problem considered by Mikhalkin in [15].
Such an enumerative problem where tropical curves have constraints imposed on their unbounded ends by a 2-form also appear in the work of T. Mandel [13]. In his paper, the author deals with the refinement of some classical invariants appearing in the balancing of some scatter-ing diagrams, thus expanding to higher dimension the refined approach of S.A. Filippini and G. Stoppa [8] to the tropical vertex group considered by M. Gross, B. Siebert and R. Pandharipande [11].
The tropical solutions to the moment problem are trivalent, and are counted with the following multiplicity: where a V and b V are two among the three outgoing slopes at the vertex V , chosen such that ω(a V , b V ) > 0. The space K ω is the space generated by the a V ∧ b V in the kernel of ω : a ∧ b ∈ Λ 2 N → ω(a, b) ∈ Z. Implicitly in the notation, the exponents of the variable q are taken modulo K ω . For a generic choice of ω, one has K ω = 0 and there is no quotient. This multiplicity is a product over the vertices of the curve, and can be seen as a Laurent polynomial in several variables, whose number depends on the rank of the quotient lattice Λ 2 N/K ω . This multiplicity also appears in [20] to prove a local invariance statement in a similar but different setting.
Remark 1.3. Getting back to the planar case, denoting by det the 2-form. The refined multiplicity B K det Γ is given by It is thus equal to the refined multiplicity of Block-Göttsche up to a factor q 1/2 − q −1/2 at each vertex and a dilatation by 2 in the exponents. It comes from the fact that q m − q −m is always divisible by q − q −1 , but in the general case, there is no such canonical choice to divide all vertex multiplicities.
We have the following invariance statement, asserting that the curve count does not depend on the choice of the hyperplanes H e .
Theorem 3.14. The count of solutions to the moment problem using the refined multiplicity B Kω Γ does not depend on the choice of H e and D as long as it is generic. It only depends on the choice of ω and the unbounded end e 0 .
In particular the curve count does not depend on the choice of D, including the choice of its slope. The obtained invariant is denoted by B ω,e 0 ∆ .
Remark 1.4. In the planar case, the refined multiplicity is the one involved in the correspondence result of Mikhalkin in [15], preferably to the Block-Göttsche multiplicity m BG Γ . As a matter of fact, denoting by N ∆ the classical refined invariant introduced in the same paper in case all points are real, one has In other terms, the multiplicity furnished by the correspondence theorem to count a tropical solution Γ and get the solutions to the classical enumerative problem tropicalizing to Γ is B K det Γ . This is no surprise since through the use of some correspondence theorem, the product over the vertices of the quantity q a V ∧b V − q −a V ∧b V carries a deep connexion to the set of all families of real oriented curves tropicalizing to Γ. This connexion stays in higher dimension.
The exponent of q here corresponds to the value of some quantum class introduced in [5], which is a generalization in higher dimension of the quantum index introduced by Mikhalkin in [15]. The existence of tropical invariants in the enumerative problems considered in the paper suggests the existence of classical refined invariants generalizing the refined invariants from [15].
Thus, we get a family of invariants B ω,e 0 ∆ ∈ Z[Λ 2 N/K ω ]. Furthermore, we prove that this map is continuous in the following sense. The map ω → K ω is constant on the cones of some fan Ω ∆ in Λ 2 M R . Denoting K σ the value on some cone σ of Ω ∆ , for τ ≺ σ, one has K σ ⊂ K τ , leading to a projection p τ We then have the following theorem. Theorem 5.3. The function ω → B ω,e 0 ∆ is constant on the interior of the cones of Ω ∆ , and for τ ≺ σ, the value on τ is the image of the value on σ under the projection p τ σ .

Main results
Theorem 3.14 is an invariance result with little improvement compared to already known tropical refined invariants since its proof relies on the fact that for the moment problem, the complex multiplicity of a tropical curve is still a product over its vertices, just as in the planar case. However, this especially simple observation can then be used to provide an invariance statement in a larger class of enumerative problems for which the complex multiplicity is no longer a product over the vertices. Consider rational tropical curves of degree ∆ in N R , and for each n e ∈ ∆, choose L e a sublattice of N/ n e such that e corkL e = |∆| + r − 3. Let L e be a affine subspace of N R / n e with slope L e . We look for rational tropical curves of degree ∆ such that image of the unbounded end e in N R / n e lies in L e . The dimension count ensures that if the L e are chosen generically, there is a finite number of solutions. Now, let ω be a 2-form such that where m C,(Le) denotes the complex multiplicity in the hereby considered enumerative problem, while m C,ω,e 0 Γ denotes the complex multiplicity in the moment problem associated to ω. See section 2.5 for a proper definition of the complex multiplicity. The choice ω = 0 always suits the hypothesis, but provides a trivial result. However, in many cases, it is possible to choose a non-trivial ω, see example 4.4. Under this technical hypothesis, we have the following invariance result.
Theorem 4.5. There exists signs ε Γ only depending on the combinatorial type of Γ and the constraints (L e ) such that the count of solutions with multiplicity ε Γ B Kω Γ only depends on the choice of the slopes L e and not the affine subspaces L e .
The obtained invariant is denoted by B Kω ∆ (L e ). The proof of the existence of the signs ε Γ is combined with the proof of invariance since the signs are in fact chosen so that the count of solutions remains invariant. Moreover, the signs ε Γ can be interpreted as orientations on the cones of the moduli space of tropical curves. With this interpretation, the multiplicities leading to invariant counts can be viewed as cycles in the star complex of the moduli space. The star complex of a fan F is the CW-complex (F\{0}) /R * + . It can be seen as the space over which F is the cone. Such an interpretation may lead to further interesting invariants.
The idea that drives the proof is that with an additional technical assumption, a multiplicity that leads to an invariant in a specific enumerative problem may be used to get a new invariant in another enumerative problem.
We also prove a statement of continuity of the invariants B Kω ∆ (L e ) in terms of the slopes (L e ). See Theorem 5.5.
Finally, we prove some enhancements of the main theorem 4.5 by imposing conditions on marked points and not only unbounded ends, and study the possibility of replacing the affine constraints by tropical cycles.

Plan of the paper
In the second section we gather the notions about tropical curves, their moduli spaces and the definition of the various involved multiplicities. The proofs for the computation of the complex multiplicities and the invariance statement it leads to are postponed to the appendices. Next, we introduce the moment problem, a specific kind of enumerative problem in higher dimension for which we prove an invariance statement. This result is used in the fourth section to provide a new invariance result in the case of larger class of tropical enumerative problems. In the fifth section we study the first properties of the new refined invariants, more precisely their dependence on the constraints of the enumerative problems, and their possible enhancements. Last but not least, we give some examples.

Abstract tropical curves
Let Γ be a finite connected graph without bivalent vertices. Let Γ 0 ∞ be the set of 1-valent vertices of Γ, and Γ = Γ\Γ 0 ∞ . We denote by Γ 0 the set of vertices of Γ, and by Γ 1 the set of edges of Γ. The non-compact edges resulting from the eviction of 1-valent vertices are called unbounded ends. The set of unbounded ends is denoted by Γ 1 ∞ , while its complement, the set of bounded edges, is denoted by Γ 1 b . An element of Γ 1 ∞ is usually denoted by e. Let l : γ ∈ Γ 1 b → |γ| ∈ R * + =]0; +∞[ be a function, called length function. It endows Γ with the structure of a metric graph by decreting that a bounded edge γ is isometric to [0; |γ|], and an unbounded end is isometric to [0; +∞[. Definition 2.1. Such a metric graph Γ is called an abstract tropical curve.
An isomorphism between two abstract tropical curves Γ and Γ ′ is an isometry Γ → Γ ′ . In particular an automorphism of Γ does not necessarily respect the labeling of the unbounded ends since it only respects the metric. Therefore, an automorphism of Γ induces a permutation of the set Γ 1 ∞ of unbounded ends. An abstract tropical curve is said to be rational if the first betti number b 1 (Γ) of the underlying graph, also called genus of the graph, is equal to 0. In graph theory, it is also called a tree.

Parametrized tropical curves
Let N be a lattice of rank r 2, and M = Hom(N, Z) be the dual lattice, which has the same rank. We denote by N R = N ⊗ Z R the associated vector space, isomorphic to R r . We now consider tropical curves inside N R .
Definition 2.2. A parametrized tropical curve in N R is a pair (Γ, h), where Γ is an abstract tropical curve and h : Γ → N R is a map such that: the restriction of h to each edge of Γ is affine with slope in N , -at each vertex V ∈ Γ 0 , one has the balancing condition: let e 1 , . . . , e k , be the edges adjacent to V , and let u 1 , . . . , u k ∈ N be the outgoing slope of h on them respectively, then one has Two parametrized tropical curves (Γ, h) and (Γ ′ , h ′ ) are isomorphic if it exists an isometry ϕ : For a parametrized tropical curve h : Γ → N R , the set of slopes of h on the unbounded ends is called the degree of the curve and is usually denoted by ∆. It is a multiset of elements of N , and due to balancing condition, one has An abstract tropical curve Γ is said to be trivalent if every vertex V ∈ Γ 0 is adjacent to precisely three edges. Let h : Γ → N R be a parametrized tropical curve with Γ trivalent. For a vertex V , let a V , b V , c V be the outgoing slopes of h, satisfying a V + b V + c V = 0. Then we define the associated bivector If a tropical curve Γ is not trivalent, we define the overvalence of the graph to be where val(V ) is the valency of a vertex. The overvalence measures the default of trivalency of the graph.
Remark 2.3. If N is of rank 2, i.e. r = 2, then π V is a multiple of a generator of Λ 2 N , which is a lattice of rank 1. The absolute value of the scalar is called vertex multiplicity.

Moduli space of rational tropical curves
We now focus on the structure of the moduli spaces of rational tropical curves, both abstract and parametrized. These spaces serve as parameter spaces for tropical curves. They are the domain of some suitable composed evaluation maps allowing us to work conveniently with the enumerative problems considered in the following sections.
Definition 2.4. Let Γ be an abstract rational tropical curve. The combinatorial type of Γ, denoted by Comb(Γ) is the homeomorphism type of the underlying graph, i.e. the labeled graph without the metric.
Conversely, to give a graph a tropical structure, one just needs to assign a length to the bounded edges of the graph. If the curve is trivalent, there are precisely |Γ 1 ∞ | − 3 bounded edges. Otherwise, there are |Γ 1 ∞ | − 3 − ov(Γ) bounded edges, where ov(Γ) is the overvalence of the graph. The set of tropical curves having the same combinatorial type is then parametrized by R  corresponds to tropical curves for which at least one edge has length 0. Those tropical curves can be identified with curves whose underlying graph has been suitably modified: edges with zero length are deleted and their extremities are merged together. These identifications allow us to glue together the cones corresponding to the finitely many combinatorial types of graphs. One obtains the moduli space M 0,|Γ 1 ∞ | of rational curves with |Γ 1 ∞ | marked points which has thus the structure of a fan.
The space M 0,m is a simplicial fan of pure dimension m − 3. The top-dimensional cones parametrize curves whose underlying graph is trivalent. Codimension 1 cones, which are called facets, correspond to combinatorial types whose underlying graph is trivalent except for one vertex which is quadrivalent. Each facet belongs to exactly three faces, which correspond to the three possible smoothings of the quadrivalent vertex into a pair of trivalent vertices.
Given an abstract rational tropical curve Γ, and a degree ∆ ⊂ N assigning a slope n e ∈ N to each unbounded end of Γ, there is a unique way up to translation to define a map h : Γ → N R which makes (Γ, h) into a parametrized tropical curve of degree ∆: the slope of the unbounded ends is fixed by the degree, and the slope of interior edges is fixed by the balancing condition. The translation is fixed by the position of a vertex in Γ. This observation gives a bijection between the moduli space M 0 (∆, N R ) of parametrized rational tropical curves of degree ∆ in N R and M 0,|∆| × N R , where the bijection is given by the projection to the underlying graph and the position of a chosen vertex, for instance the vertex adjacent to a fixed unbounded end e 0 .
The cones of the moduli space M 0 (∆, N R ) are of the form R |∆|−3 + × N R , whose tangent space contains the lattice Z |∆|−3 × N .
Remark 2.5. The multiset ∆ is allowed to contain the 0 vector. Unbounded ends on which the map h has zero slope are sometimes called marked points because their image inside N R is a contracted to a point.
Remark 2.6. Notice that the set of unbounded ends of parametrized tropical curves in M 0 (∆, N R ) are canonically identified since they are labeled. Thus, one can speak about Γ 1 ∞ without having to specify a curve Γ. The degree ∆ is then a map e ∈ Γ 1 ∞ → n e ∈ N .
The set of combinatorial types of parametrized tropical curves of degree ∆ is denoted C ∆ , where the C stands both for combinatorial and for cones. Among them, the set of top-dimensional combinatorial types is denoted by C top ∆ . Otherwise, the set of cones of a given dimension is denoted by C (k) ∆ . A combinatorial type is usually denoted by the letter C, whose notation is also used to denote the associated cone in the moduli space M 0 (∆, N R ). Moreover, for a given parametrized tropical curve h : Γ → N R , we sometimes forget the h in the notation, and use the same notation Γ for the curve and the combinatorial type Comb(Γ) that it belongs to.
At some point, we need to orient the cones of the moduli space M 0 (∆, N R ), which means find an orientation for the underlying lattice Z |∆|−3 × N of each cone C ∈ C top ∆ , or similarly, its linear span. If C ∈ C top ∆ is a combinatorial type, associated to an orthant of the moduli space M 0 (∆, N R ) and denoted by the same letter, we denote by or(C) an orientation of the linear span of the cone. If W is a facet adjacent to C, W belongs to the boundary of C. Then, as the normal bundle of W in C is canonically oriented, an orientation or(C) induces an orientation of the tangent bundle to W , which is denoted by ∂ W or(C). Notice that a reversing of the orientation of C also reverses the induced orientation on W .

Evaluation map
For each unbounded end e ∈ Γ 1 ∞ , let n e be the slope assigned by the degree ∆. One has a welldefined evaluation map on the moduli space M 0 (∆, N R ), which sends a parametrized tropical curve to the image of the unbounded end inside the quotient N R / n e . If the slope is 0, this quotient is just N R itself and the image is a point, justifying the name marked point. Otherwise, the image corresponds to the line that contains the image of the unbounded end under h. Putting all these maps together, we get the evaluation map: where p e is any point chosen in e ∈ Γ 1 ∞ .
By definition, the evaluation map is linear on every cone of M 0 (∆, N R ) ≃ M 0,|∆| × N R but is not necessarily injective on top-dimensional cones. For instance, it is not the case when some bounded edge has zero slope, or when the combinatorial type contains a flat vertex, which is a vertex whose adjacent edges are contained in a line.

Complex multiplicity of tropical curves
One main feature of tropical enumerative geometry is to count tropical curves solution to a given enumerative problem P, which means finding the curves which are subject to some constraints. It is often necessary to use suitable multiplicities m to get a count that does not depend on the choice of the constraints when those vary in some family. Such a count using multiplicity m is said to lead to an invariant for the problem P. Here, we describe the main multiplicities that are of interest to us, procrastinating the proper introduction to the associated enumerative problems.
Remark 2.7. In some particular enumerative problems, some multiplicities leading to an invariant are provided through the use of some correspondence theorems, such as Mikhalkin's [14], Shustin's [19], Nishinou and Siebert's [17] or Tyomkin's [22]. The use of these multiplicities leads to tropical invariants which have the value of some classical invariants, e.g. Gromov-Witten invariants, Welschinger invariants, or refined invariants introduced by Mikhalkin in [15]. Such a relation to classical invariants is not systematic although is sometimes suspected, as it is the case for refined invariants introduced by Block and Göttsche [3].
In this section, we recall the definition of the complex multiplicity, which is the multiplicity that comes from the correspondence theorem [14] in the planar case, or [17] in the higher dimensional case. Let Γ be a parametrized tropical curve of degree ∆ ⊂ N . For each unbounded end e, let L e be a full sublattice of N/ n e , meaning that the quotient is without torsion. In order to get small notations, we denote by Q e the quotient of N/ n e by L e , which is a lattice of rank rk N/ n e − rk L e = cork L e . We denote by Q R e = Q e ⊗ Z R the associated vector space. We assume that e∈Γ 1 ∞ corkL e is equal to the dimension of M 0 (∆, N R ), which is |∆| + r − 3. We then compose the evaluation map with the quotient maps N R / n e → Q e , getting what we call a composed evaluation map: For a combinatorial type C ∈ C top ∆ , we denote by ev (Le) C the restriction to the orthant of the combinatorial type C, which is linear. Due to the assumption on the ranks of the lattices, both spaces have the same dimension.
Definition 2.8. We set the complex multiplicity to be where the domain is endowed with a basis of the lattice Z |∆|−3 × N , and the codomain a basis of the lattice e Q e . Remark 2.9. One other way to describe the complex multiplicity is as follows: consider the sublattice e L e ⊂ e N/ n e , which is a full sublattice of codimension |∆|+r−3 by assumption. Let α (Le) ∈ Λ |∆|+r−3 ( e N/ n e ) * be its Plücker vector obtain as follows: take the wedge product of a basis of the dual orthogonal of e L e . Then the pull-back of α (Le) by the evaluation map ev C restricted to a cone C is a top-dimensional form. Thus, it is a scalar multiple of the determinant form on the lattice of the orthant. The absolute value of the scalar factor is the multiplicity.
It also follows from standard tropical intersection theory (see [1]) that the complex multiplicity corresponds to the index of the lattice spanned by ev C (Z |∆+r−3 ×N ) and e L e inside e N/ n e , which are of complementary dimension if ev is injective on the cone relative to the combinatorial type. If not, the complex multiplicity is 0.
The complex multiplicity can be computed in the hereby described fancy way. To keep a general setting, here and only here, we consider tropical curves (Γ, h) that are not necessarily trivalent. Nevertheless, for such a curve and an evaluation map the complex multiplicity is still defined as the determinant of ev (Le) restricted to the cone of the combinatorial type, provided that it has the same dimension as the target space. We assume it is the case.
For each primitive sublattice L ⊂ N of corank l, there is an orthogonal dual primitive sublattice L ⊥ ⊂ M of rank l, and an associated Plücker vector ρ ∈ Λ l M defined up to sign. This polyvector is defined as follows: let m 1 , . . . , m l be a basis of L ⊥ . Then, one has This polyvector does not depend on the chosen basis up to sign. Hence, given a parametrized rational tropical curve h : Γ → N R , one has a polyvector ρ e associated to each of the unbounded ends e of the curve: the Plücker vector associated to the full sublattice spanned by n e and L e .
We choose one vertex of Γ to be the sink of the curve and orient every edge toward it. We then cut the branches of the tropical curve with the following rule: let V be a vertex different from the sink, with incoming unbounded ends directed by n 1 , . . . , n s , and respective polyvectors ρ 1 , . . . , ρ s , and a unique outgoing bounded edge, thus directed by n 1 + · · · + n s . The polyvector associated to the outgoing edge of V is ρ = ι n 1 +···+ns (ρ 1 ∧ · · · ∧ ρ s ).
Recall that ι n , for n ∈ N , denotes the interior product by n. Geometrically, the polyvector associated to an edge and to an unbounded end is a multiple of the Plücker vector associated to the space described by the edge when it moves. This means the following: -For an unbounded end e ∈ Γ 1 ∞ associated to a sublattice L e , the unbounded end can move in the direction L e , thus describe the space with slope spanned by n e and L e .
-At a vertex V with incoming edges having respective polyvectors ρ 1 , . . . , ρ s . Assume by induction that the incoming edges move in an affine space directed by a vector subspace whose Plücker vector is ρ i respectively. Then, the vertex moves in the intersection of all these subspaces. Therefore, the affine subspace has Plücker vector ρ 1 ∧ · · · ∧ ρ s .
-Finally, for the outgoing edge of V , it moves in an affine space equal to the affine space where V lives, enlarged by the direction of the edge: n 1 + · · · + n s . Hence, the polyvector is obtained by making the interior product with n 1 + · · · + n s .
At the sink, let ρ 1 , . . . , ρ s be the polyvectors associated to the incoming adjacent edges. Because of the assumption on dimensions, we have Thus, it is an integer multiple of a generator of Λ r M . The absolute value of the proportionality constant, obtained by evaluating on a basis of N , is the desired determinant. The proof that it does not depend on the choice of the sink and that it computes the determinant is proved in the appendix.
Theorem 6.8. The value obtained by the preceding algorithm is equal to the complex multiplicity m C Γ .

Refined multiplicities of a trivalent tropical curve
To finish this section about tropical curves, we define the refined multiplicities that we use in the following sections. Let V ∆ be the set of types of trivalent vertices that occur in some combinatorial type of M 0 (∆, N R ). A trivalent vertex is determined by the slopes of the three outgoing edges. Using the balancing condition, the type of a vertex is fully determined by the partition of ∆ into three subsets that it induces. Hence, V ∆ is finite.
Let ω 0 be a 2-form such that for any V ∈ V ∆ with π V = 0, ω 0 (π V ) = 0. This condition means that for any non-flat vertex, the restriction of ω 0 to the plane spanned by the outgoing edges is non-zero. The refined multiplicity is defined as follows.
Definition 2.10. The refined multiplicity of a parametrized tropical curve h : where the bivector π V of V is chosen such that ω 0 (π V ) > 0. If the curve contains some flat vertex W , then π W = 0 and then B 0 Γ = 0.
The refined multiplicity is a Laurent polynomial in several variables. The number of indeterminates is equal to the rank of Λ 2 N , which is Å r 2 ã . It depends up to sign on the choice of ω 0 .
Let K ⊂ Λ 2 N be a sublattice. Then we can compose the definition of B 0 Γ with the projection p K : Z[Λ 2 N ] −→ Z[Λ 2 N/K] to get a slightly less refined multiplicity.
, obtained by reducing the exponents modulo K.
Notice that this notation is consistent with the notation for B 0 Γ , which corresponds to the choice K = {0}. Although the relation p K (B 0 Γ ) = B K Γ shows that B 0 Γ determines the other refined multiplicities, the use of these multiplicities is necessary since the multiplicity B 0 Γ might fail to provide an invariant, while some B K Γ might do so. Remark 2.12. If N is of rank 2, then Λ 2 N ≃ Z, and the choice of a generator of Λ 2 N identifies we recover the definition of the refined multiplicity proposed by Block and Göttsche in [3].
Remark 2.13. The refined multiplicity depends up to a sign on the choice of ω 0 . In every occurrence of the refined multiplicity, the choice of ω 0 is fixed in a non-ambiguous way so it does not bring up any problem.

Moment problem and first refinement
Before turning to more general enumerative problems and providing them with a refined way to count the solutions, we present a family of enumerative problems, called moment problems, where it is slightly easy to show the existence of an associated refined invariant, i.e. a refined way to count the solutions of the problem and leading to a result independent of the constraints.
In the planar case, meaning that r = 2, one can consider the classic enumerative problem of finding degree ∆ rational tropical curves passing through |∆|−1 points. In that case, the complex multiplicities of rational tropical curves split themselves as a product over the vertices of the curve. This fact, which is specific to the two-dimensional case and fails in arbitrary dimension, is used to prove invariance with the refined multiplicity of Block-Göttsche [3], see [12]. The moment problems also share this property of the complex multiplicity splitting into a product over the vertices of the curve.
Remark 3.1. In the planar case, the moment problem amounts to find rational tropical curves passing through |∆| − 1 points when these points are located on the toric divisors, meaning that the unbounded ends are contained in fixed lines of the same slope.

Enumerative problem
Let ∆ ⊂ N be a tropical degree consisting only of non-zero vectors, and let ω ∈ Λ 2 M be a 2-form chosen such that ω(n e , −) = 0 for every n e ∈ ∆. Before getting to the enumerative problem, let us define what we mean by moment. This definition may be extended to the moment of a bounded edge of Γ if an orientation is specified. Geometrically, the moment corresponds to the position of the affine space H e of slope H e = ker ι ne ω that contains the image unbounded end e. Using the balancing condition at each vertex of the curve, one easily proves the following proposition. Remark 3.4. This fact generalizes the planar Menelaus relation noticed by Mikhalkin in [15].
That is why we also call it tropical Menelaus relation.
We now state the moment problem. Consider the global moment map The moment map, which has image in M due to the tropical Menelaus relation, is a specific case of composed evaluation map, where one chooses L e = H e .
Problem 3.5. Let µ ∈ H. How many parametrized rational tropical curves of degree ∆ satisfy mom(Γ, h) = µ ? In other words, for each e ∈ Γ 1 ∞ \{e 0 }, fix H e an affine hyperplane with slope H e . How many parametrized rational tropical curves of degree ∆ satisfy h(e) ⊂ H e ?
Unfortunately, the codomain has dimension |∆|−1, while the domain has dimension |∆|+r−3. Therefore, unless r = 2, this is not enough to ensure a finite number of solutions. We then add the following condition: let e 0 ∈ Γ 1 ∞ be a specific unbounded end and let δ ∈ N/ n e 0 be a primitive vector such that ω(n e 0 , δ) = 0 and D be a line with slope δ. We add the condition that h(e 0 ) ∈ D ⊂ N R / n e 0 , getting to the moment problem denoted by P(ω, e 0 ). Problem 3.6. Let µ ∈ H and D be a line with slope δ inside N R / n e 0 . How many parametrized rational tropical curves of degree ∆ satisfy mom(Γ, h) = µ and h(e 0 ) ∈ D ?
Remark 3.7. The condition h(e 0 ) ∈ D means that the image of the unbounded end e 0 is contained in a plane.
We denote the associated composed evaluation map as follows: where the R terms correspond to the evaluation of the moments of the unbounded ends different from e 0 .
Remark 3.8. We omit δ from the notation P(ω, e 0 ), since, as proven in Theorem 3.14, the solutions do not depend on δ. -The constraint could have been split into two constraints on two different unbounded ends.
-One could have impose constraints on one or several new marked points on the curve. Some of these other possibilities are studied later on in section 5.

Count of solutions and complex invariance statement
We now get to the invariance statement for the count of solutions to P(ω, e 0 ) using first the complex multiplicity, meaning that the count does not depend on the choice of µ nor D as long as it is generic. The invariance of the count using complex multiplicity has already been proven in various ways: by hand showing the local invariance around walls, using tropical intersection theory, or using the invariance in the complex setting and some correspondence theorem. In any case, the statement is a particular case of a more general statement which is proved in the appendix. Yet, as the invariance with the complex multiplicity is a key ingredient in the proof of invariance using refined multiplicity, we give here the main steps of what could be called the pedestrian proof. Proof. The set of top-dimensional combinatorial types C top ∆ decomposes between the types on which the composed evaluation map ev ω,e 0 is non-injective, and the ones where it is bijective. The image of the cones where the composed evaluation map is non-injective is nowhere dense inside the codomain since ev ω,e 0 is linear and non-surjective either. The same applies for the image of the cones which are not top-dimensional. Therefore, if (µ, D) is chosen outside the image of these cones, the only cones that may provide a solution are top-dimensional ones where the evaluation map is injective. Such a solution, if it exists, is unique. As there are finitely many combinatorial types, the conclusion follows.
Remark 3.11. This fact can also be proven using tropical intersection theory: for a generic choice of constraints, the spaces ev(M 0 (∆, N R )) and D × e =e 0 H e intersect transversally inside e N R / n e and intersection points belong to the interior of top-dimensional faces. The conclusion follows along with the invariance of the count using complex multiplicity.
We now compute the complex multiplicities of the solutions.
Lemma 3.12. For each trivalent solution h : Γ → N R to the moment problem, the complex multiplicity takes the following form: Proof. We use the algorithm to compute the complex multiplicity by choosing as sink the vertex adjacent to the unbounded end e 0 . Then, the recursion uses the following identity: Hence, the computation reduces to the case there is just one vertex. Noticing that the constraint on e 0 is associated to the polyvector ι δ ι ne 0 Ω, where Ω is a generator of Λ r M , we have To conclude, for a generic choice (µ, D), let be the count of solutions to P(ω, e 0 ) using the complex multiplicity. This count is well-defined since the solutions Γ are trivalent. Then one has the following invariance statement. The proof follows from tropical intersection theory, or from the computations done in the appendix.

Refined invariance statement
We conclude this section by showing an invariance statement for the same enumerative problem 3.6, using this time the refined multiplicities B Kω introduced in section 2.6.
be the space spanned by the vertices spanning a plane where ω restricts to 0. If ω is chosen generically regarding ∆, then K ω = 0. Otherwise, the 2-form ω might take value 0 on some vertex V occurring in a rational tropical curve of degree ∆. We use the refined multiplicity B Kω Γ of a tropical curve, which is The 2-vector π V is chosen so that ω(π V ) > 0. If ω(π V ) = 0, then π V ∈ K ω and the multiplicity is equal to 0 since the exponents are taken modulo K ω . Then, for a generic choice (µ, D), let Proof. We already know that the count of solutions with complex multiplicity m C,ω,e 0 leads to an invariant. This means that the repartition of the solutions around a wall matches the invariance of the count with complex multiplicities. Recall that here, a wall is a tropical curve with a unique quadrivalent vertex. Therefore, we need to check for each wall that the count with refined multiplicities is also invariant.
We check the local invariance around the wall corresponding to a quadrivalent vertex of the tropical curve. Let a 1 , a 2 , a 3 , a 4 be the slopes of the outer edges, with index taken in Z/4Z. Up to a relabeling, we assume that ω(a i , a i+1 ) > 0 for every i. If some value is equal to 0, the proof remains unchanged. Moreover, we assume that ω(a 2 , a 3 ) > ω(a 1 , a 2 ). The invariance of the complex count amounts to the following relation: ω(a 1 , a 2 )ω(a 1 + a 2 , a 3 ) + ω(a 1 , a 3 )ω(a 2 , a 1 + a 3 ) + ω(a 2 , a 3 )ω(a 2 + a 3 , a 1 ) = 0, for 12//34 for 13//24 for 14//23 and the repartition of combinatorial types around the wall is given by the sign of each term. It means up to sign that one is positive and is on one side of the wall, and the two other ones are negative, on the other side of the wall. Hence, we just need to study the signs of each term to know which curve is on which side. We know that ω(a 1 , a 2 ) and ω(a 1 + a 2 , a 3 ) = ω(a 3 , a 4 ) are positive. Therefore, their product, which is the term of 12//34, is also positive. We know that ω(a 2 , a 3 ) is positive, but ω(a 2 + a 3 , a 1 ) = −ω(a 4 , a 1 ) is negative. Therefore, their product is negative and 14//23 is on the other side of the wall. It means that the combinatorial types 12//34 and 14//23 are on opposite sides of the wall. We need to determine on which side the type 13//24 is, and that is given by the sign of the middle term. As by assumption ω(a 2 , a 1 + a 3 ) = ω(a 2 , a 3 ) − ω(a 1 , a 2 ) > 0, it is determined by the sign of ω (a 1 , a 3 ).
-If ω(a 1 , a 3 ) > 0, then 12//34 and 13//24 are on the same side, and the invariance for refined multiplicities is dealt with the identity and if ω(a 1 , a 3 ) < 0, then 14//23 and 13//24 are on the same side and then the invariance for refined multiplicities is true since (q a 2 ∧a 3 − q a 3 ∧a 2 )(q a 1 ∧(a 2 +a 3 ) − q (a 2 +a 3 )∧a 1 ) This closes the proof if each complex multiplicity is non-zero. If the multiplicity of some side of the wall is 0, it does not provide any solution to the enumerative problem for generic values, and one needs to cancel its refined multiplicity, which might not be 0 if one chooses the refined multiplicity B 0 Γ . If some combinatorial type has complex multiplicity 0, it means that for some W among the two displayed vertices, ω(π W ) = 0. The same relations between refined multiplicity lead to the invariance provided that one mods out the exponent by π W , which sends the refined multiplicity to 0. One just obtains that the curves on the two remaining sides of the wall have the same multiplicity.
The result also does not depend on δ since the refined multiplicity does not depend on δ, and the set of solutions for a given choice of (µ, D) only depends on µ and the intersection point of D with the hyperplane H e 0 .
Remark 3.15. We have proven a refined invariance statement for every ω such that for every n ∈ ∆, ι n ω = 0. For every ω not taking the value 0 on V ∆ , we have an refined invariant in Z[Λ 2 N ]. If for some V one has ω(π V ) = 0, the space of the exponents where the refined invariant takes value is reduced by π V .

Refined count for rational tropical curves
We now use the refined invariance proven in the setting of the moment problem to prove a refined invariance for a larger class of enumerative problems, where the constraints we ask a tropical curve to satisfy do not necessarily come from the choice of a 2-form ω.

New enumerative problems
Let ∆ ⊂ N be a tropical degree without any zero vector. For each n e ∈ ∆, let L e be a full sublattice of N/ n e , and let L e be an affine subspace of N R / n e with slope L e . A full sublattice means that the quotient is without torsion. Assume that e corkL e = |∆| + r − 3. In other words, using previous notations, solving this enumerative problem means finding the preimages of a generic point for the following composed evaluation map The enumerative problem is referred as P(L e ), and is related to an analogous complex enumerative problem though the use of a correspondence theorem. The count with the complex multiplicity m C,(Le) Γ is already known to lead to an invariant. The result is recalled here is proven in the appendix. Furthermore, the result of this count is equal to the complex invariant provided by the analogous complex problem. However, the involved complex multiplicity m C,(Le) does not in general split into a product over the vertices of the curve. Nevertheless, we prove that for a suitable choice of K and up to a change of sign, the refined multiplicity B K Γ , which is a product over the vertices, does lead to a invariant. Meanwhile, let us describe the combinatorial types that have complex multiplicity 0. (iv) The curve has no flat vertex or contracted edge, the ranks of constraints are rightly dispatched but non-generic.
Proof. In the first two cases, the multiplicity is zero since the evaluation map is non-injective, so neither is the composed evaluation map. Let us focus on the third case and assume there is a bounded edge γ ∈ Γ 1 b such that the constraints are wrongly dispatched. Out of symmetry, let assume that e∈A cork L e > r + |A| − 2 and the sink used to compute the multiplicity is in the B part of the curve. The recursive formula ρ = ι n+n ′ (ρ 1 ∧ ρ 2 ) used to compute the complex multiplicity shows that the polyvector ρ γ obtained applying the algorithm is the interior product with a polyvector belonging to Λ A corkLe−|A|+2 M . By assumption, this space is 0 since the weight is strictly bigger than r, and so is the multiplicity. By disjunction, the last case contains indeed the remaining cases. Using the Plücker embedding, the space of corank l sublattices of a given lattice N is Λ l M . As Hom(N/ n e , Z) ≃ n e ⊥ ⊂ M , the complex multiplicity can be seen as a multilinear form on the space e Λ corkLe n e ⊥ of possible slopes for the lattices L e , once their dimensions are fixed. The assumption ensures that the multilinear form itself is non-zero, but it might still take the zero value on some specific choice of slopes. This is avoided if the slopes are chosen generically.

Main results
For a choice of slope constraints (L e ), let ω ∈ Λ 2 M be a 2-form satisfying the following property: for every top-dimensional combinatorial type Γ, In particular, if for some combinatorial type Γ which is not of type (i) or (ii) one has m C,(Le) Γ = 0, then K ω = {0}. Such a ω may not be unique and in some cases might even be 0. However, the choice of ω does not really matter, only the space K ω matters. Example 4.4. Choose the boundary constraints on every unbounded end to be at least codimension 1 and having generic slope, and let ω be a 2-form such that K ω = {0}, meaning that the only combinatorial types with complex multiplicity relative to ω equal to 0 are the curves with a flat vertex or a contracted edge. Then, we prove that the complex multiplicity relative to (L e ) is never 0 unless there is a flat vertex or contracted edge.
According to the classification of Lemma 4.2, we can only be in the case (iii). Thus, we show that the dimensions are rightly dispatched. Let γ ∈ Γ 1 b be an edge and A⊔B = Γ 1 ∞ the associated partition. Then by assumption, We now get to the main result. We use the refined multiplicity B Kω Γ , where the bivectors π V are chosen such that ω(π V ) > 0. If ω(π V ) = 0, then B Kω Γ = 0 and the sign does not matter.
Theorem 4.5. There exists a collection of signs ε Γ indexed by the combinatorial types C top ∆ such that the following refined count does not depend on the choice of the constraints L e as long as it is generic. It only depends on the choice of L e .
Remark 4.6. It might happen that for some combinatorial type m C,(Le) Γ = 0 and still B Kω Γ = 0, but not the other way around.
The construction of the collection of signs is given in the proofs. They can be obtained as follows. Let C be a top-dimensional combinatorial type in C top ∆ with non-zero complex multiplicity m C,(Le) C . We have two evaluation maps restricted to the cone C: In other words, if one identifies the codomains with R |∆|+r−3 and one chooses an orientation on it, the sign ε C compares the orientations on C which are induced by both composed evaluation maps.

Proofs of invariance
We give two different proofs of the result. Both proofs use a trick that might be of interest in other situations as well: without giving to much context, given two tropical enumerative problems P and P ′ for tropical curves having the same walls, assuming some additional condition on the combinatorial types not providing any solution, it is possible to use a multiplicity m leading to an invariant in P to get an invariant in P ′ , eventually twisting the multiplicity m with a sign. In our case, we use the invariance with respect to the refined multiplicity B Kω in the moment problem P(ω, e 0 ) to get an invariance in the problem P(L e ).
First proof of Theorem 4.5. The first way of proving the existence of the signs is choosing the sign of one combinatorial type and spreading the signs so that one has local invariance around the walls, which are here the curves with a quadrivalent vertex. Then, one just has to check that the signs are well-defined.
Let W ∆ be the set of walls, i.e. combinatorial types of curves with a quadrivalent vertex. Consider the graph on C top ∆ where the two combinatorial types C and C ′ are adjacent if they are adjacent to a common wall W ∈ W ∆ . We consider the subgraph G reg ⊂ C top ∆ only containing the combinatorial types with non-zero multiplicity B Kω C , and thus non-zero complex multiplicity m C,(Le) C , along with the edges between them. This graph might not be connected anymore. For a given connected component, let C 0 be a fixed combinatorial type, thus having non-zero complex multiplicity m C,(Le) C 0 . Let ε C 0 = 1. For a given wall W ∈ W ∆ , we write the local invariance relation for B Kω around W in the following way: where η W C are equal to ±1, and depend on the repartition of the sides of the wall in the enumerative problem P(ω, e 0 ). The triple (η W C 1 , η W C 2 , η W C 3 ) is defined up to global change of sign. If for a given combinatorial type C one has B Kω C = 0, the signs η • C are not uniquely defined but their definition is unimportant. Now, for a combinatorial type C ∈ G reg ⊂ C top ∆ , let be a path going from C 0 to C in G reg , where W i ∈ W ∆ is a wall adjacent to both combinatorial types C i and C i+1 . Now, we use the following rule to define the signs ε C i : let C ′ i ∈ C top ∆ be the last combinatorial type adjacent to W i , so that the adjacent combinatorial types are C i , C i+1 and C ′ i . We are in one of the following situations: -The combinatorial types C i and C i+1 are on the same side of the wall, so that the invariance relation for the complex multiplicity m C,(Le) writes itself We write the invariance relation for B Kω mimicking the previous relation, and multiply it so that the coefficient of m Kω Thus, we choose -If C i and C i+1 are on different sides of the walls, the invariance for the complex multiplicity writes itself We then proceed similarly and choose are on different sides of the wall, and the combinatorial type C ′ i never provides any solution to the enumerative problem for a generic value. As by assumption B Kω C ′ i is also 0, the previous case still works.
For each connected component of G reg , the choice of a sign on one combinatorial type C 0 thus propagates to define a sign for every other combinatorial type in the connected component. The rule to propagate the sign from one side of the wall to another might be reformulated in the following closed way. Fix an orientation on the codomain of the composed evaluation map ev (Le) . Let or(C) be the orientation of C induced ev (Le) C . For a combinatorial type on which the composed evaluation map is not invertible, the orientation is chosen at random and does not matter. Then, two combinatorial types C i and C i+1 stand on the same side of the wall W i if and only if the induced orientations ∂ W i or(C i ) and ∂ W i or(C i+1 ) on W i coincide. Thus, we can use the closed formula We now check that the signs are well-defined, meaning that the definition of ε C does not depend on the chosen path between C 0 and C. Notice that the rule of propagation is reversible: the sign of C i determines the sign of C i+1 in the same way that the sign of C i+1 determines the sign C i . Thus, we only have take a loop from C 0 to itself, and prove that the sign obtained by the propagation rule is equal to the original sign ε C 0 = +1. Let be such a loop inside a connected component of G reg . Then, the iteration of the rule of signs gives the following sign on C 0 : and needs to be equal to 1. This can be written as a condition on the signs η: Splitting the right-hand side product and reindexing its numerator, we get to Notice that this condition does not depend on the up to local sign choice of the η, and does neither depend on the η • C for C with B Kω C = 0 since the loop stays inside a component of G reg . As each factor of the right-hand side is unchanged if the orientation of a combinatorial type C i is reversed, then the condition does not depend on the orientations of each C. Therefore, the rule to propagate signs depends on the orientations or(C) chosen on the combinatorial types, but the condition to ensure that these signs are uniquely defined only depends on the η W C , not on the chosen orientations. Thus, we can choose any orientation to check the formula, for instance, the orientation induced by the composed evaluation map ev ω,e 0 , related to P(ω, e 0 ).
As the refined multiplicity B Kω is known to give an invariant in the problem P(ω, e 0 ), the local invariance around a wall W i gives by definition η W C = ∂ W orω(C) or(W ) , for a fixed orientation or(W ) of W . Thus, It implies that the relation is verified, which concludes the proof since by construction the local invariance is satisfied at each wall.
Remark 4.7. We could have noticed from the start that the η W C are given by the formula ∂ W orω(C) or (W ) and conclude that the necessary relation to guarantee that signs were well-defined is satisfied. However, this wouldn't emphasize the fact the condition solely depends on the signs in front on the multiplicities in the dependence relations given by the walls.
This first proof gives the existence of the signs but fails to provide a useful definition, which is given in the following second proof, whose idea consists in interpreting the multiplicities leading to an invariant as cycles on the moduli space M 0 (∆, N R ).
Second proof of Theorem 4.5. Consider the orientation or(C) of C ∈ C top ∆ induced by the composed evaluation map ev (Le) and a fixed orientation of its codomain. If the restriction to a given combinatorial type is not invertible, the chosen orientation does not matter. We then consider the following top dimensional chain Technically, this is a chain in the star complex of the fan M 0 (∆, N R ), obtained as the quotient of M 0 (∆, N R )\{0} by R * + . The fan M 0 (∆, N R ) is the cone over its star complex, which is compact a CW-complex. Nevertheless, we write M 0 (∆, N R ) to avoid further notations which are not essential to the proof, denoting the cells of the star complex by their associated cone in M 0 (∆, N R ). Choosing an arbitrary orientation or(W ) of every wall, the boundary of Ξ is where C 1 , C 2 and C 3 are the sides adjacent to the wall W , and the sign η W C i = ∂ W or(C) or(W ) is given by the boundary map. Then, the local invariance relation needed to show that multiplicities m (Le) lead to an invariant for the enumerative problem P(L e ) is equivalent to the fact that Ξ is a cycle: ∂Ξ = 0. Conversely, any m = (m C ) C such that  Remark 4.9. The second proof gives an explicit description of signs. However, the first proof might provide several possible signs: one choice of signs for each connected component of the graph G reg . Recall that G reg is the graph of combinatorial types whose multiplicity is non-zero. It would be an interesting question to know whether this graph is always connected or not. If the graph were not connected, the sum of multiplicities of curves solution to an enumerative problem and whose combinatorial type belongs to a given component of G reg would lead to refined invariants of a new kind.

First properties and possible enhancements of refined invariants
We now focus ourselves on some properties of these new refined invariants, including their relation to the complex multiplicities. Then, we provide some directions to enlarge and improve, or not, the family of refined invariants.

Relation to the complex multiplicities
We here prove a relation between the refined and the complex multiplicities in the case of the moment problem P(ω, e 0 ). In the general setting, to some extent with prove the non-existence of such a relation. The spirit of this section is to find a way to recover m C Γ from B K Γ .
Let us first consider the planar setting, i.e. r = 2, and let det be a generator of Λ 2 M . Recall that the refined multiplicity of Block-Göttsche is defined as follows: for a trivalent parametrized tropical curve h : Γ → N R , where π V is chosen so that det(π V ) > 0. This refined multiplicity is related to our choice of refined multiplicity through the relation ] is the morphisms that evaluates det/2 on the exponents. As Γ , the refined multiplicity determines the complex multiplicity in the planar case, and so the refined invariant B ω,e 0 ∆ determines the complex invariant N ω,e 0 ∆ .
Getting back to the general case, this limit relation extends to any dimension, and thus proves that the refined multiplicity determines the complex multiplicity in the moment problem P(ω, e 0 ) for any ω, which is stated in the following proposition.
, and thus In the general case of the enumerative problem P(L e ), the question remains to know whether the complex multiplicity can be recovered using the refined multiplicity and some clever function involving the constraints. Sadly, the complex multiplicity cannot be determined solely with the refined multiplicity.
Remark 5.2. It seems possible to find tropical curves satisfying constraints of the same slope, and with the same refined multiplicity but different complex multiplicities, preventing from finding a simple relation between the refined multiplicity and the complex multiplicity.

Regularity of the invariants
For a fixed degree ∆, the invariants B ω,e 0 ∆ and B K ∆ (L e ) form a large family of invariants depending on ω for the first family, and (L e ) for the second The 2-form ω varies in the space Λ 2 M ⊗ R, while (L e ) varies in e Λ le n e ⊥ , where l e = corkL e . In this section we study the dependence of these invariants in their respective parameters.
The case of the invariants B ω,e 0 ∆ is dealt with in the following theorem. First, let us introduce the fan structure Ω ∆ on Λ 2 M R , which is induced by the linear forms ω → ω(π V ), for every V ∈ V ∆ . Notice that the function ω → K ω = π V : ω(π V ) = 0 V ∈V ∆ is constant on the cones of the subdivision, and thus denoted by K σ for a given cone σ of Ω ∆ . Furthermore, if τ ≺ σ are cones, then K σ ⊂ K τ . The top-dimensional cones are those for which K σ = {0}. Remember that B ω,e 0 ∆ is defined whenever ∆ ∩ {n : ι n ω = 0} = ∅. The set of ω not satisfying this assumption is some subfan of Ω ∆ , where one can set B ω,e 0 ∆ = 0. under the projection Proof. Let ω 0 be a fixed 2-form where B ∆ is defined. Let C ∈ C top ∆ be a combinatorial type with non-zero multiplicity m C,ω 0 ,e 0 C and Γ 0 be a curve belonging to the interior of C. We consider ev ω,e 0 as a function of both ω and Γ ∈ C. This way, the partial differential ∂ev ω,e 0 ∂Γ at (ω 0 , Γ 0 ) is the usual differential of ev ω 0 ,e 0 , which is itself since it is linear on C. Therefore, the differential is invertible for its determinant is equal to the complex multiplicity m C,ω 0 ,e 0 Γ 0 , non-zero by assumption. Thus, the implicit function theorem ensures that at the neighborhood (ω 0 , Γ 0 ), the equation ev ω,e 0 (Γ) = ev ω 0 ,e 0 (Γ 0 ) in (ω, Γ) solves itself with Γ being a function Γ(ω), with Γ(ω 0 ) = Γ 0 .
Let us use the previous fact to prove the theorem. For a generic value (µ, D), at ω 0 , one has For the combinatorial types that contribute a solution in the above sum, the previous fact using implicit functions theorem shows that they still provide a solution when ω moves in the neighborhood of ω 0 . For the same reasons, combinatorial types where the composed evaluation map is invertible but which do not provide a solution continue not to provide a solution. The remaining combinatorial types never provide a solution for P(ω 0 , e 0 ), but might start to do so for some P(ω, e 0 ). However, their multiplicity B Kω 0 Γ is 0. Moreover, the function is constant in a neighborhood of ω 0 : if the value is 0, it stays zero since some π V belongs to K ω 0 , otherwise the conditions ω(π V ) > 0 remains valid for ω in a neighborhood of ω 0 . In particular, for ω belonging to the same cone of Ω ∆ , since then one has K ω = K ω 0 , the multiplicity stays the same, and ω → B ω,e 0 ∆ is locally constant for ω in the neighborhood of ω 0 and in the same cone. The first statement follows since cones are connected.
If ω is still in the neighborhood of ω 0 but does not belong to the same cone of Ω ∆ , then the same observation applies. And B ω,e 0 ∆ = B ω 0 ,e 0 ∆ ∈ Z[Λ 2 N/K ω 0 ] which means that B ω 0 ,e 0 ∆ is the image of B ω,e 0 ∆ under the map that projects the exponents Z[Λ 2 N/K ω ] → Z[Λ 2 N/K ω 0 ], thus proving the second statement.
Remark 5.4. Clearly, the moment problem P(ω, e 0 ) depends on ω only up to multiplication by a scalar. The fan Ω ∆ induces a subdivision on the projective space P(Λ 2 M R ), which is a hyperplane arrangement. Then, [ω] → B ω,e 0 ∆ is a function constant on the cells of the subdivision and satisfying the compatibility condition when going from a cell to one of its faces. On the cells where B ω,e 0 ∆ is not defined, the map can be extended by 0. This choice is natural since for such 2-forms, the composed evaluation map is never surjective, and there is an invariant which has value 0.
For the general enumerative problem P(L e ), the same reasoning applies. Let (L 0 e ) be a specific set of slopes constraints. Let l e = corkL 0 e , so that the tuple (L e ) varies in L = e Λ le n e ⊥ . is continuous on L. Thus, in a neighborhood of (L 0 e ), one still has m C,(Le) C = 0. Thus, the 2-form ω also satisfies the condition for (L e ) and thus provides an invariant in Z[Λ 2 N/K ω ]. The implicit functions theorem provides the local invariance in the same way as for the moment problem case: combinatorial types where the complex multiplicity is non-zero provide or not a solution for a specific value of the constraints, and we apply the implicit functions theorem, and combinatorial type where the complex multiplicity cancels might provide a solution for (L e ) in a neighborhood of (L 0 e ) but their multiplicity B Kω C is 0 by assumption.

Tropical cycles constraints
We now try to enlarge the scope of the refined invariance theorems by generalizing the family of enumerative problems P(L e ) involved in Theorem 4.5. The first direction in which one could look is to replace the constraints, which are for now affine subspaces in N R / n e , by tropical cycles of the same dimension inside the same space. Let (Ξ e ) be a family of tropical cycles Ξ e ⊂ N R / n e such that e codimΞ e = |∆| + r − 3.
Problem 5.6. The problem referred as P(Ξ e ) is counting rational tropical curves of degree ∆ which intersect Ξ e for every e ∈ Γ 1 ∞ .
The dimension count ensures that if the cycles Ξ e are chosen generically, there is a finite number of solutions. In this situation, general theory still provides a complex multiplicity which leads to an invariant. However, the proof of Theorem 4.5 for the refined invariance fails in this context for the following reasons.
-First, in the case of affine constraints, each tropical curve h : Γ → N R determines uniquely the constraints that it satisfies: L e is the affine space with slope L e passing through h(e) ∈ N R / n e . This can be reprased as the existence of the composed evaluation map M 0 (∆, N R ) → e Q R e , where Q R e is a moduli space for the affine subspaces L e of slope L e . The proof of invariance then consists in showing that this map as a well-defined degree for various multiplicities. This is not the case for tropical cycle constraints (Ξ e ) since they vary in a wider moduli space: there is no composed evaluation map anymore. Thus, there is no suitable definitions of the "walls" inside M 0 (∆, N R ) that could lead to an analogous proof.
-Then, if one tries instead to mimic the invariance proof for the complex multiplicity, whether it uses tropical intersection theory or not, we have to check invariance around one new kind of wall: where the intersection point between one tropical curve Γ solution to the enumerative problem and one of the constraints Ξ e belongs to a facet of Ξ e . This intersection is not generic among the choices of Ξ e , and the invariance for the complex multiplicity is provided by the balancing condition of Ξ e , which would express as ρ 1 + ρ 2 + ρ 3 = 0 for the Plücker vectors of the adjacent facets of Ξ e . However, at this wall, two solutions on one side of the wall become one on the other side, but the combinatorial types of the solutions are the same. This prevents any local invariance using a multiplicity solely depending on the combinatorial type of the curve, as is B K Γ . -Despite the impossibility of the multiplicity B K Γ to provide a local refined invariant, it might exists a clever way to find another refined multiplicity leading to an invariant. However, such a multiplicity would not have the close relationship that B 0 Γ shares with the count of real curves tropicalizing to Γ through the correspondence theorem.
Nevertheless, we can prove refined invariance in the following particular case. Let Ξ e 0 ⊂ N R / e 0 be a tropical cycle of dimension 1, i.e. a tropical curve, and ω ∈ Λ 2 M be a 2-form. We consider the following variant of the moment problem.
Problem 5.7. How many rational tropical curves h : Γ → N R of degree ∆ have fixed family of moments µ ω and satisfy h(e 0 ) ∈ Ξ e 0 ?
We count the solutions to the problem with multiplicity |ω(n e 0 , δ)|B Kω Γ , where δ is the slope of Ξ e 0 at the intersection point with Γ: We have the following invariance result.
Theorem 5.8. For a generic choice of (µ ω , Ξ e 0 ), the count does not depend on the choice of µ ω nor the choice of Ξ e 0 . It only depends on ω and the degree of Ξ e 0 . Moreover, its value is where ω · Ξ e 0 is the intersection index between Ξ e 0 and an affine hyperplane of slope H e 0 .
Proof. The proof is immediate because due to Theorem 3.14, for each intersection point between H e 0 and Ξ e 0 , the refined count is |ω(n e 0 , δ)|B ω,e 0 ∆ , the result follows by definition of the intersection index.

Constraints on marked points
The other main direction to generalize the refined invariance theorems by staying in the realm of cycles constraints would be to allow the constraints to live in the main strata N R . This would mean to allow unbounded ends with slope 0 and impose cycle constraints on them.
The first problem that occurs is that the moment problem, which is at the core of the proof of invariance, does not allow unbounded ends to have slope 0. If we are to go by the same proof as for Theorem 4.5, the first step is then to find an suitable enumerative problem that does so. For instance, the following variant of the moment problem. The idea is to add marked points that do not add any constraints for the curves.
Let ϕ 1 , . . . , ϕ p ∈ M be p non-zero linear forms, and let H i ⊂ N R be affine hyperplanes with respective slope ker ϕ i . Let as usual be ω a 2-form, µ ∈ M a family of moments and D ⊂ N R / e 0 , all chosen generically. We consider the following enumerative problem P (ω, e 0 , (ϕ i )). We consider curves of degree ∆ ⊔ {0 p }, which means that curves have now p new unbounded ends of slope 0, which are called marked points and are denoted by i ∈ [[1; p]].
Problem 5.9. P (ω, e 0 , (ϕ i )) How many parametrized rational tropical curves h : Γ → N R of degree ∆ ⊔ {0 p } satisfy mom(Γ) = µ, h(e 0 ) ∈ D, and h(i) ∈ H i ? Remark 5.10. The new additional conditions are kind of void for the following reason. Forgetting the marked points, each solution of P (ω, e 0 , (ϕ i )) gives a solution of the usual moment problem P(ω, e 0 ). Conversely, every solution to P (ω, e 0 , (ϕ i )) is obtained from a solution to the standard moment problem P(ω, e 0 ) by choosing intersection points with the hyperplanes H i . Such points always exist. Therefore, the solutions are morally the same solutions as for the moment problem. The new marked points on the curve just have to be chosen at the intersection points between the solutions to the moment problem and H i .
Standard intersection theory ensures that the count of solutions with complex multiplicity yields an invariant. Using the algorithm to compute the complex multiplicity of a solution h : Γ → N R , one sees that the vertex corresponding to the marked point i just contributes a factor |ϕ i (δ Γ i )|, where δ Γ i is the slope of Γ at the marked point. Hence, the complex multiplicity tales the following form: Using Remark5.10, one obtains that by adding all the multiplicities over the curves solutions to problem 5.9, where ϕ i · ∆ = n∈∆:ϕ i (n)>0 ϕ i (n) denotes the intersection index between a curve of degree ∆ and a hyperplane of slope ker ϕ i . The combinatorial types where ϕ i (δ C i ) have thus zero complex multiplicity. They were previously part of the fourth type. They do not occur if (ϕ i ) are chosen generically regarding ∆.
We now get to the refined count of this enhanced moment problem.
Proposition 5.11. The count of solutions to problem P (ω, e 0 , (ϕ i )) with refined multiplicity yields an invariant which is equal to ( i ϕ i · ∆) B ω,e 0 ∆ . Proof. The proof uses Remark 5.10 which describes solutions to problem 5.9 in terms of solutions to the usual moment problem. The direct computations yields the result.
We can then proceed, applying the same trick as in the proof of Theorem 4.5, to get a new invariance result. Let L e ⊂ N/ n e for e ∈ Γ 1 ∞ and L i ⊂ N i for i ∈ [[1; p]] be full sublattices such that Let L e , L i be affine subspaces of respective slopes L e , L i . We consider the following enumerative problem. Theorem 5.14. There exists some signs ε Γ such that the refined count of solutions to problem 5.12 using multiplicity ε C i |ϕ i (δ C i )|B Kω C leads to an invariant. Proof. The proof is identical to the proof of Theorem 4.5.
Remark 5.15. One gets many new invariants depending on the choice of the linear forms ϕ i . Clearly, the invariants are 1-homogeneous in the linear forms ϕ i . Yet, it would be interesting to study the various obtained invariants.
As before, there is a subtle condition for the existence of refined invariants, which is the existence of ω, (ϕ i ). To see that this generalization is still of interest, notice that the class of enumerative problems for which constraints on the unbounded ends and marked points are of codimension at least 1 provide a family of examples for which a generic choice of ω and ϕ i satisfy the assumption, leading to refined invariants in Z[Λ 2 N ].
As in the case of tropical cycle constraints, the appearance of factors i |ϕ i (δ C i )| makes the new multiplicity lose its relationship to the real curves tropicalizing to the tropical curve.

Examples of refined invariants
We now give a few examples, limited for the two following computational technicalities: first, we do not have yet a way of solving efficiently either of the enumerative problems P(ω, e 0 ) and P(L e ), secondly, in the case of P(L e ), the finding of a 2-form ω might be difficult. Example 6.1. First, let assume that ω = ϕ ∧ ψ is a generic 2-form of rank 1, where ϕ, ψ ∈ M . Then, the 2-form has a (r − 2)-dimensional kernel ker ω = ker ϕ ∩ ker ψ which is contained in every ker ι ne ω. Therefore, the translate of a curve h : Γ → N R by a vector in ker ω has the same moments. Hence, solving the moment problem in N R can be done by solving the moment problem in N R / ker ω and then lifting the solutions to N R : a solution h : Γ → N R to P(ω, e 0 ) gives a solution to P(ω) to the moment problem in N R / ker ω by composing with the quotient map N R → N R / ker ω, which is endowed with the quotient form ω. Conversely, any solution in the quotient can uniquely be lifted to a solution in N R such that h(e 0 ) belongs to the line D chosen for the enumerative problem. Therefore, we reduce to the planar case and the computation can be done using the recursive formula from [4].
ij ] 1 i<j 4 through its canonical basis: q ij = q e i ∧e j .
In particular, one can see that although B it is immediate, and for B ω − ,e 1 ∆ , the first two rows give the value of B ω 0 ,e 1 ∆ while the last rows cancels when q 12 goes to 1. ♦ Example 6.5. We now give an example of refined invariant in the general case. We consider the enumerative problem with the following constraints: = q 12 q 13 q 14 q 23 q 24 q 34 − q 34 q 12 q 13 q 14 q 23 q 24 − q 12 q 13 q 14 q 24 q 34 q 23 + q 23 q 12 q 13 q 14 q 24 q 34 + q 12 q 13 q 14 q 34 q 23 q 24 − q 23 q 24 q 12 q 13 q 14 q 34 − q 12 q 13 q 23 q 14 q 24 q 34 + q 14 q 24 q 34 q 12 q 13 q 23 − q 12 q 14 q 34 q 13 q 23 q 24 + q 13 q 23 q 24 q 12 q 14 q 34 + q 12 q 34 q 13 q 14 q 23 q 24 − q 13 q 14 q 23 q 24 q 12 q 34 .
♦ Example 6.6. Last, we give an example where a generic choice of ω does not satisfy the assumption of Theorem 4.5. We consider the enumerative problem of lines passing through two points which are located on the unbounded ends directed by e 1 and e 5 . Notice that by standard geometry, there is always exactly one line passing through two points. Hence, the complex multiplicity is either 0 or 1, and combinatorial type can provide a solution only one at a time.
The combinatorial type where some of the unbounded ends directed by e 2 , e 3 and e 4 are adjacent to a common vertex have complex multiplicity 0. One can check that the other combinatorial types, of the form 1i//j//k5 have complex multiplicity 1. This notation denotes the combinatorial type where unbounded ends directed by e 1 and e i are adjacent to the same vertex, as well as e k and e 5 . In order to apply Theorem 4.5, we can choose a 2-form ω that cancels on e 2 ∧ e 3 , e 2 ∧ e 4 and e 3 ∧ e 4 . For instance, take Recall that the term of each product are obtained as q π V − q −π V where ω 4 (π V ) > 0. For instance, for the midle vertex of the combinatorial type 13//2//45, one has ω 4 ((e 1 + e 3 ) ∧ e 2 ) = 1 > 0, thus, q (e 1 +e 3 )∧e 2 = q e 1 ∧e 2 q −e 2 ∧e 3 = q 12 q 23 .

Appendix: computation of the complex multiplicity
We here prove that the algorithm described in section 2.5 computes the complex multiplicity.
Lemma 6.7. The value of the multiple does not depend on the chosen sink.
Proof. We prove that the obtained value does not change if we replace the sink V by one of its neighbors. Let V be a vertex, W one of its neighbors, and E the edge between them, directed by n. Let ρ 1 , . . . , ρ s be the polyvectors of the edges adjacent to V different from E, and ρ ′ 1 , . . . , ρ ′ s ′ the polyvectors associated to the edges adjacent to W different from E. The computation leads to the two following results: -If V is the sink, we get Therefore, the equality (up to sign) comes from the fact that ρ 1 ∧ · · · ∧ ρ s ∧ ρ ′ 1 ∧ · · · ∧ ρ ′ s ′ is 0 since it is in Λ r+1 M = {0}, and that ι n is a derivation on Λ • M .
Theorem 6.8. The value obtained by the preceding algorithm is equal to the complex multiplicity m C Γ .
Proof. We make an induction on the number of vertices of the curve Γ, and cut the branches one by one.
-If the curve has just one vertex, let ρ i = m i1 ∧ · · · ∧ m ir i be the polyvectors associated to the lattices L i for the s unbounded edges of the curve. The evaluation matrix of ev (Le) , denoted by [f ], has the following form: Therefore, we have det ev (Le) = m 11 ∧ · · · ∧ m 1r 1 ∧ · · · ∧ m srs = ρ 1 ∧ · · · ∧ ρ s .
-If Γ has more than one vertex, let V be a vertex adjacent to only unbounded ends associated to polyvectors ρ 1 , . . . , ρ s , and one unique neighbor vertex W . We keep the same notations ρ i = m i1 ∧ · · · ∧ m ir i . We choose a basis of the cone associated to the combinatorial type of Γ consisting of the canonical basis of R m−3−ov(Γ) 0 , and the N R factor corresponding to the position of W . Then, the matrix of ev (Le) has the following form: where the first columns correspond to the evaluation of the vertex W , the last column to the evaluation corresponding to the edge between V and W . We make a development with respect to the last column. We get the following result: (−1) • m ij , n * · · · * * · · · * . . . . . . . . . . . . . . . . . . * · · · * * · · · * m 11 0 · · · 0 . . . . . . . . . . . .
Each determinant in the sum is the determinant of the evaluation matrix ev (Le) for a tropical curve where the vertex V is deleted and the edge between V and W replaced by an unbounded end, associated with a constraint having polyvector (−1) • m ij , n m 11 ∧ · · · ∧ m ij ∧ · · · ∧ m srs . The sum of these vectors is precisely ι n (ρ 1 ∧ · · · ∧ ρ s ). Hence, the result follows by induction.

Appendix: invariance for the complex multiplicity
We consider parametrized rational tropical curves of degree ∆ in N R . We allow vectors of ∆ to be 0, so that the unbounded ends associated to these vectors are marked points on the curves. For each unbounded end e, let L e be a primitive sublattice of N/ n e , and let l e be its corank.
We denote by L R e = L e ⊗ R. Notice that if e corresponds to a marked point, L e is a sublattice of N . Let L e be a generic affine space in N R / n i with slope L R e . This amounts to the choice of a point λ e in N R / n e ⊕ L R e . We prove the invariance of the solutions to problem P(L e ) using the complex multiplicity m C,(Le) . Lemma 6.9. For a generic choice of (λ e ), there is a finite number of solutions to P(L e ). Moreover, these solutions are trivalent.
Proof. There is a finite number of cones in the moduli space M 0 (∆, N R ). Moreover, on each cone, i.e. each combinatorial type, ev (Le) is linear. If the restriction of ev (Le) to this cone is injective, the combinatorial type contributes at most one solution.
The point (λ e ) can always be chosen outside the image of the non top-dimensional cones, since these images are included in proper subspaces of e Q R e . For a combinatorial type of topdimension, as by assumption its dimension is equal to the dimension of e Q R e , if ev (Le) is not injective, it is not surjective either, and its image is thus contained in a proper subspace. Finally, a generic choice of (λ e ) is chosen outside the image of the non top-dimensional cones and the cones where ev (Le) is not injective.
For such a choice of (λ e ), there is a finite number of solutions, and the corresponding curves are trivalent since they belong to top dimensional cones of M 0 (∆, N R ). Now, let (λ e ) be chosen generically. We set N ∆ (L e ) = ev (Le) (Γ,h)=(λe) m C Γ .
Notice that if the subspaces L e are chosen generically, there is no curve with zero multiplicity that contributes to the sum, otherwise ev (Le) is not injective nor surjective on the cone corresponding to the combinatorial type of the curve, and thus (λ e ) has been chosen out of its image.
Theorem 6.10. The value of N ∆ (L e ) only depends on the slopes L e of the affine subspaces and not their specific choice as long as it is generic.
This invariant is thus denoted by N ∆ (L e ).
Proof. As many proofs of tropical invariance, the proof goes by the study of local invariance at the walls of the tropical moduli space. The proof is similar to Proposition 4.4 in [?]. We proceed in two steps: first, we show that the sum of the determinants of the composed evaluation maps around a wall is zero, and then we show that the sign of these determinants characterizes the existence of solutions, thus proving the local invariance.
• We consider the wall associated to a quadrivalent vertex in the tropical curve. Let the adjacent edges be denoted by the indices 1, 2, 3, 4. The three adjacent combinatorial types are determined by the splitting of the quadrivalent vertex into two trivalent vertices. These possibilities are denoted by 12//34, 13//24 and 14//23. The cone in the moduli space M 0 (N R , ∆) corresponding to each combinatorial type is the quadrant of the vector space N R × R m−4 × R, consisting of the points with positive coordinates on the R entries, where the N R factor corresponds to the vertex V adjacent to the edge 1, the R m−4 has canonical basis indexed by the edges of the curve with the quadrivalent vertex, which are common to all curves in the adjacent combinatorial types, and the R factor corresponds to the length of the edge resulting from the splitting of the quadrivalent vertex.
Let v j be a directing vector of the edge j, oriented outward the quadrivalent vertex. For each marked point or unbounded end i, associated to a constraint L i , let m i1 , . . . , m ir i be linear forms defining the sublattice L i : L i = r i j=1 ker m ij . Then, for the combinatorial type 12//34, the matrix of the composed evaluation map ev (Le) takes the following form: .
We make the sum of the three determinants for the three adjacent combinatorial types, and use the linearity with respect to the last column, since all the other columns are equal. We get Using a combination of the columns corresponding to N R applied to v 1 , and the balancing condition v 1 + v 2 + v 3 + v 4 = 0, we get . Now, we see that the last column is the sum of the columns indexed 1, 2, 3, 4. Thus, the sum of the determinants is 0. If some of the edges 1, 2, 3, 4 was unbounded, the columns with the corresponding indices would not appear, but for the linear forms m ij that would be evaluated on the corresponding unbounded end, one would already have m ij , v i = 0 and the result is unchanged. Finally, one has detA 12//34 + detA 13//24 + detA 14//23 = 0.
• We now use the previous statement to prove the invariance of the count. We denote by L the tuple (L e ). Let L(t) be a generic path between two generic configurations L(0) and L(1), i.e. a path in e Q R e . Outside a finite set of values of t, the configuration L(t) is generic and N ∆ (L(t)) is given by a sum over some combinatorial types with non-zero multiplicity. More precisely, on each combinatorial type with non-zero multiplicity, the composed evaluation map ev (Le) is a linear map associated with a matrix A. If the coordinates on the combina-torial type are denoted by (V, l), the equation can be formally solved: (V, l) = A −1 L, and this provide a true solution of ev (Le) (Γ, h) = L if the coordinates of l are non-negative.
As the multiplicity only depends on the combinatorial type, the value of N ∆ (L(t)) is locally constant at generic L(t), thus, we only need to show invariance around the special values t where L(t) is not generic. Let t * be such a value. At least one of the curves of (ev (Le) ) −1 (t * ) has a quadrivalent vertex, and it deforms into the adjacent combinatorial types when t moves slightly around t * . Let A 12//34 , A 13//24 and A 14//23 be the matrices of ev (Le) on the three adjacent combinatorial types. On each combinatorial type, we can solve uniquely A ⋆ (V, l) = L(t): (V, l) = A −1 ⋆ λ(t), where ⋆ is one of the three adjacent combinatorial types, and we get a true solution if all the coordinates of l are non-negative. This is the case for all the edges except the edge that appears with the splitting of the quadrivalent vertex. Using Cramer's rule to solve A ⋆ (V, l) = L(t), the length of the new edge is equal to detÃ⋆ detA⋆ , whereÃ ⋆ is the matrix A ⋆ with the last column (the one that corresponds to the length of the new edge) being replaced with L ∈ i Q R e . As the matrices A 12//34 , A 13//24 and A 14//23 only differ in their last column, the numerators are all equal, and the sign of the length of the new edge is thus determined by the sign of detA ⋆ . Finally, the sign of the determinant determines which combinatorial type provide a true solution, and the local invariance follows from the relation detA 12//34 + detA 13//24 + detA 14//23 = 0.
Remark 6.11. The invariance also results from general results of tropical intersection theory [1]. Had we worked with more general tropical cycles Ξ e , the proof with intersection theory would also work, but in that case, we would have more walls to study. These walls correspond to the cases where the parametrized tropical curves do not intersect the cycles Ξ e in their top-dimensional faces. The invariance would then result from the balancing condition for the cycles Ξ e .