Introduction to tropical series and wave dynamic on them

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

1.1. Motivation and history. The purpose of this article is two-fold: we develop the theory of a dynamic on tropical series on two-dimensional domains and prove ancillary statements for our works about sandpile dynamic (see [10], [11] for details), non-commutative toric varieties (see [22] for a sketch), and number π (see [8]). Our initial motivation was [5] where it was experimentally observed that tropical curves appear in sandpile models and behave nicely when we add more sand. We think that this text has an independent interest, so we separated it from [10] to make it more accessible for a general audience.
We experimentally observed, [7], that the dynamic on the space of tropical series, proposed here, obeys power law. Namely, the distribution of the area of an avalanche (a direct analog of that for sandpiles) in this model has the density function of the type p(x) = cx α . To the best of our knowledge, this simple geometric dynamic is the only model, among the ways to obtain power laws in a simulation, which produces a continuous random variable.
For a general introduction to tropical geometry, see [4], [20], or [3]. It seems that the results of this paper can be extended to higher dimensions, but the proofs will be technically more involved.
1.2. Plan of this paper, main objects. A tropical series on a closed convex domain Ω ⊂ R 2 is just a function which is locally a minimum of a finite number of functions ix + jy + a ij where i, j ∈ Z, a ij ∈ R. For the purpose of [10] we need to consider tropical series which are non-negative. Clearly, nonnegative tropical series on R 2 are just constants, so we restrict our attention to admissible Ω's, see Definition 2.8. We study general properties of tropical series in Sections 2, 3. Somewhat the main tropical series associated to Ω is the weighted distance function, see Section 4.

INTRODUCTION TO TROPICAL SERIES AND WAVE DYNAMIC ON THEM
In Section 5 we define the main character of this paper, the "wave" operator G p , where p ∈ Ω • and G p acts on tropical series, and study its properties. The word "wave" stands for the fact that G p is the scaling limit incarnation of sending waves from p in a sandpile model, see [10] for details. Section 6 is devoted to the dynamic generated by applying operators G p for different points. In Section 7 we show how to lift G p to an operator on Laurent polynomials over a field of characteristic two. Its algebraic meaning is yet to be discovered.
In Sections 9, 10 we show how to approximate Ω by a somewhat canonical family of Q-polygons, i.e. (possibly non-compact) polygons with finite number of sides of rational slopes. Sections 12,13 further reduce the study of dynamic to the case of so called nice tropical series, which behave well near the boundary of Ω. Proposition 12.4 tells that in order to approximate G = G p1 G p2 . . . 0 Ω , by so-called blow-ups we can restrict the dynamic to a Q-polygon close to Ω. Proposition 13.2 asserts that by changing G just a bit we may assume that all tropical curves are smooth (Definition 8.1) during this dynamic. We summarize these results in Section 15.
The tropical curve C(G) for the above function G has the minimal tropical symplectic area (defined in [24]) among the curves passing through the points p 1 , . . . , p n , see Section 14 where we also explain the name of this notion.
Smooth tropical curves corresponding to nice tropical series are main objects in the proofs in [10], [11]. Using results of this paper we will reduce the theorems in [10] to the local questions which can be addressed purely combinatorially with help of super-harmonic functions [9].
1.3. Acknowledgments. We thank Andrea Sportiello for sharing his insights on perturbative regimes of the Abelian sandpile model which was the starting point of our work on sandpiles. Our proofs required developing the theory of tropical series, presented here.
The first author, Nikita Kalinin, is funded by SNCF PostDoc.Mobility grant 168647. Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.
The second author, Mikhail Shkolnikov, is supported in part by the grant 159240 of the Swiss National Science Foundation as well as by the National Center of Competence in Research SwissMAP of the Swiss National Science Foundation.

Tropical series
Recall that a tropical Laurent polynomial (later just tropical polynomial) f on U ⊂ R 2 in two variables is a function f : U → R which can be written as where A is a finite subset of Z 2 . Each point (i, j) ∈ A corresponds to a monomial ix + jy + a ij , the number a ij is called the coefficient of f of the monomial corresponding to the point (i, j) ∈ A. The locus of the points where a tropical polynomial f is not smooth is a tropical curve (see [20]). We denote this locus by C(f ) ⊂ U . Example 2.4. Tropical Θ-divisors [21] are tropical analytic curves in R 2 , as well as the standard grid -the union of all horizontal and vertical lines passing through lattice points, i.e. the set The following example illustrates that a tropical series on Ω • in general cannot be extended to ∂Ω. For all tropical series f with C(f ) = C, the sequence of values of f (x, y) tends to −∞ as x → 0.
Question 2.6. When can we extend a tropical series from Ω • to ∂Ω?
Tropical series on non-convex domains exhibit the behavior as in the following example.
Example 2.7. The function f (x, y) = min (3, x + [y]) is a tropical series on the following U : x + 2) and the monomial x appears with different coefficients 0, 2 in the different parts of U .
Definition 2.8. (Used on pages [1,6,13]) A convex closed subset Ω ⊂ R 2 is said to be not admissible if one of the following cases takes place: • Ω has empty interior Ω • (i.e. Ω is a subset of a line), • Ω is R 2 , • Ω is a half-plane with the boundary of irrational slope, • Ω is a strip between two parallel lines of irrational slope. Otherwise, Ω is called admissible. Definition 2.9. (Used on pages [4,14] Note that l ij Ω is positive on Ω • . Also, A Ω always contains (0, 0). Proof. It is easy to verify that if Ω is not admissible, then Ω • = ∅ or A Ω = {(0, 0)}. Let us prove "only if" direction. Since Ω = R 2 , there exists a boundary point z of Ω and a support line l at z. If the slope of l is rational, then A Ω contains the corresponding lattice point; if this slope is irrational but l does not belong to the boundary of Ω, then there exists another support line of Ω with a close rational slope. So, we may suppose that l is contained in the boundary of Ω. If there is no other boundary points of Ω, then Ω is a half-plane and is not admissible. If there exists another boundary point in ∂Ω \ l, then we repeat the above arguments and find a support line of Ω with rational slope. Another case, i.e. that Ω is a strip between two lines of the same irrational slope, is not possible since Ω is admissible.

Ω-tropical series
From now on we always suppose that Ω is an admissible convex closed subset of R 2 . Definition 3.1. (Used on pages [2,4,4]) An Ω-tropical series is a function f : and A ⊂ Z 2 is not necessary finite. An Ω-tropical analytic curve C(f ) on Ω • is the corner locus (i.e. the set of non-smooth points) of an Ω-tropical series on Ω • .
The reason to consider only admissible sets is Proposition 2.10 is that an Ω-tropical analytic curve on non-admissible Ω is always the empty set, because either Ω • is empty or the only Ω-tropical series is the function 0. An Ω-tropical series can be thought of an analog of a series f t (x, y) = (i,j)∈AΩ t aij x i y j with t ∈ R >0 very small. Is is true that Ω is the limit of the images of the region of convergence of f t under the map log t : (x, y) → (log t |x|, log t |y|), and the corresponding Ω-tropical analytic curve is the limit of the images of {f t (x, y) = 0} under log t | · | when t → 0? Lemma 3.4. (Used on pages [4,4,7,8,13,13]) Let U ⊂ R 2 be an open set and K ⊂ U be a compact set. For any C > 0 the set i.e. the set of monomials which potentially can contribute on K to an Ω-tropical function f with max K f ≤ C, is finite.
Lemma 3.5. (Used on pages [5]) In the definition of an Ω-tropical series f , (3.2), we can replace "inf" by "min", i.e. at every point (x, y) ∈ Ω • we have Proof. Suppose that for a point (x 0 , y 0 ) ∈ Ω • and for each (i, j) ∈ A the value of the monomial a ij + ix 0 + jy 0 is distinct from the value of the infimum inf (i,j)∈A Thus, there exists C > 0 such that we have a ij + ix 0 + jy 0 < C for infinite number of monomials (i, j) ∈ A. Since (a ij + ix + jy)| Ω ≥ 0 for all (i, j) ∈ A, applying Lemma 3.4 yields a contradiction.
At a point on ∂Ω where there is no tangent line with a rational slope we actually have to take the infimum, cf. the proof of Lemma 4.4. Similarly, applying Lemma 3.4 for small compact neighbors of points we obtain the following result. Note that an Ω-tropical series f : Ω → R always has different presentations as the minimum of linear functions. For example, if Ω is the square [0, 1] × [0, 1] ⊂ R 2 , then min(x, 1 − x, y, 1 − y, 1/3) equals at every point of Ω to min(x, 1 − x, y, 1 − y, 1/3, 2x, 5 − 2x). Definition 3.7 (cf. [13], Lemma 5.3). (Used on pages [7]) To resolve this ambiguity, we suppose that, in Ω • , a tropical series f is always (if the opposite is not stated explicitly) given by (Used on pages [4,7,8] with A = A Ω (Definition 2.9) and with as minimal as possible coefficients a ij . We call this presentation the canonical form of a tropical series. For each Ω-tropical series there exists a unique canonical form. Proof. It is easy to check that f (x, y) = min(x, 1 − x, y, 1 − y, 1/3) on Ω. All the coefficients a ij , (i, j) = (0, 0) are chosen as minimal with the condition that ix + jy + a ij is non-negative on Ω. Finally, in the canonical form of min(x, 1 − x, y, 1 − y, 1/3) the coefficient a 00 can not be less than 1/3. Proof. Let f | U = ix + jy + a ij for an open U ⊂ Ω • . It follows from convexity of Ω and local concaivity of f that f (x, y) ≤ ix + jy + a ij on Ω. Therefore in Ω • we have

Tropical distance function
Definition 4.1. (Used on pages [5,5,7]) We use the notation of (2.9). The weighted distance function l Ω on Ω is defined by l Ω (x, y) = inf l ij Ω (x, y)|(i, j) ∈ A Ω \ {(0, 0)} . An example of a tropical analytical curve defined by l Ω is drawn on the right hand side of Figure 1.
The same argument in the proof of Lemma 3.5 proves the following lemma.    [4,7]) The function l Ω is an Ω-tropical series.
Proof. It is enough to prove that l Ω is zero on ∂Ω and continuous when we approach ∂Ω. It is clear that l Ω = 0 on the points of {l ij Ω = 0} ∩ ∂Ω for all (i, j) ∈ A Ω . Suppose that there exists a point (x 1 , y 1 ) = z ∈ ∂Ω where the only support line L is of irrational slope α.
Since Ω is admissible, there exists a point z = (x 2 , y 2 ) ∈ L which does not belong to ∂Ω. Using continued fractions for α, we get two sequences of numbers, p 2n /q 2n < α < p 2n+1 /q 2n+1 such that |α − p m /q m | < 1/q 2 m , for all m, and q m tends to infinity. Either for all even k, or for all odd k the line through z with the slope p k /q k does not intersect Ω, so (−p k , q k ) ∈ A Ω or (p k , −q k ) ∈ A Ω . Thus, for such k the absolute value of the linear function which tends to zero as i → ∞. Therefore, we can construct a sequence of functions l k , k → ∞, whose values at z tend to zero, and l Ω ≤ |l k |. This proves both continuity of l Ω at z and that l Ω (z) = 0 for all z ∈ ∂Ω.

Wave operators G p
Recall that Ω ⊂ R 2 is admissible (Definition 2.8). Let P = {p 1 , . . . , p n } be a finite collection of points in Ω • . Let g be an Ω-tropical series.
Definition 5.1. (Used on pages ) Denote by V (Ω, P, f ) the set of Ω-tropical series g such that g| Ω ≥ f and g is not smooth at each of the points p ∈ P. Proof. Since Ω is admissible, l Ω is well defined, and the function Definition 5.3. (Used on pages ) For a finite subset P of Ω • and an Ω-tropical series f we define an operator G P , given by Lemma 5.4. (Used on pages [8]) Let g and f be two tropical series on Ω • such that g ≤ f and In Lemma 5.9 we prove that each individual G p simply contracts a face of a tropical curve C(f ) until C(G p f ) passes through p, see Figure 4. In Proposition 6.1 we will prove that G P can be obtained as the limit of repetitive applications G p for p ∈ P .
We denote by 0 Ω the function f ≡ 0 on Ω.
• Figure 3. On the left: Ω-tropical series min(x, y, 1−x, 1−y, 1/3) and the corresponding tropical curve. On the right: the result of applying G ( 1 1 3 ) and the corresponding tropical curve is presented on the right. The fat point is ( 1 5 , 1 2 ). Note that there appears a new face where 2x is the dominating monomial.
Proof. Indeed, all the coefficients, except a 00 , in the canonical form of G p 0 Ω can not be less than in l Ω by Remark 4.2, and if a 00 were less than l Ω (p), then the function would be smooth at p. Proposition 5.6. (Used on pages [7,8,11]) For any z ∈ Ω and P = {p 1 , . . . , p n } the following inequality holds Proof. For each point p ∈ P we consider the function (G p 0 Ω )(z) = min(l Ω (z), l Ω (p)), which is not smooth at p and (G p 0 Ω )| ∂Ω = 0. Finally, Lemma 5.7. (Used on pages [12]) If f is an Ω-tropical series, then G P f is an Ω-tropical series. verified The set M is finite by Lemma 3.4. Therefore, the restriction of any tropical series g ∈ V (Ω, P, f ) to K can be expressed as a tropical polynomial min (i,j)∈M (ix+jy+a ij (g)). In particular, if we denote by a ij the infimum of a ij (g) for all g ∈ V (Ω, P, f ) then so G P f is a tropical series. It follows from Proposition 5.6, that G P f ≤ f + n · l Ω . Then, l Ω | ∂Ω = 0 by Lemma 4.4. Therefore G P f | ∂Ω = 0 and, thus, Lemma 3.10 concludes the proof that G P f is an Ω-tropical series.
Lemma 5.9. (Used on pages [6,8,9]) Let f = min (i,j)∈AΩ (ix + jy + a ij ) be an Ω-tropical series in the canonical form, suppose that p = (x 0 , y 0 ) ∈ Ω • \ C(f ). Suppose that f is equal to kx + ly + a kl near p. Consider the function f (x, y) = min Then, Figure 4. Illustration for Remark 5.12. The operator G p shrinks the face Φ where p belongs to. Firstly, t = 0, then t = 0.5, and finally t = 1 in Add ct ij f . Note that combinatorics of the curve can change when t goes from 0 to 1.
by definition. Therefore f and G p f differ only at one monomial. Also, direct calculation shows that min(f , kx + ly + c) is smooth at p as long as c < f (p) − kx 0 − ly 0 , which finishes the proof.
Corollary 5.11. In the notation of Definition 4.1, for a point p ∈ Ω • , for each z ∈ Ω we have 6. Dynamic generated by G p for p ∈ P .
. . } be an infinite sequence of points in P where each point p i , i = 1, . . . , n appears infinite number of times. Let f be any Ω-tropical series. Consider a sequence of Ω-tropical series {f m } ∞ m=1 defined recursively as Proposition 6.1. The sequence {f m } ∞ m=1 uniformly converges to G P f . Proof. First of all, G P f has an upper bound f + nl Ω by arguments as in Proposition 5.6. Applying Lemma 5.4, induction on m and the obvious fact that G pm G P f = G P f we have that f m ≤ G P f for all m. It follows from Lemmata 3.4, 5.9 that G qm , m = 1, . . . change only a certain fixed finite subset of monomials in f m . This implies the uniform convergence: since the family {f m } ∞ m=1 is pointwise monotone and bounded, it converges to some Ω-tropical seriesf ≤ G P f . Indeed, to find the canonical form off we can take the limits (as m → ∞) of the coefficients for f m in their canonical forms (3.8).
It is clear thatf is not smooth at all the points P . Therefore, by definition of G P we havef ≥ G P f , which finishes the proof. Remark 6.2. Note that in the case when Ω is a lattice polygon and the points P are lattice points, all the increments c of the coefficients in G p = Add c kl are integers, and therefore the sequence {f m } always stabilizes after a finite number of steps. Lemma 6.3. (Used on pages [9]) Let ε > 0, B ⊂ Z 2 and f, g be two tropical series in Ω • written as If |δ ij | < ε for each (i, j) ∈ B, then C(f ) is 2ε-close to C(g).
Proof. Let z ∈ C(f ), l 1 , l 2 be two monomials of f , which are minimal at z. Suppose that B 2ε (z)∩C(g) = ∅. Therefore g| B2ε(z) = l(x, y) where l : R 2 → R is a linear function with integer slope. Without loss of generality we may suppose that z = (0, 0), f (z) = 0 and l(x, y) = c. Clearly, c ≥ −ε. At least one of l 1 , l 2 is not a constant, by SL(2, Z)-change of coordinates we may suppose that l 1 = x. Then, in g, the monomial x has the coefficient δ 1,0 which satisfies |δ 1,0 | ≤ ε. But then x + δ 1,0 ≤ −ε ≤ c at a point in B 2ε ((0, 0)), so this point belongs to C(g), which is a contradiction.
Remark 6.4. Note that if G qn . . . G q1 f is close to the limit G P f , then by Lemma 6.3 we see that the corresponding tropical curves are also close to each other.
Proof. Note that for each z ∈ Ω, |f (z) − g(z)| ≤ ρ(f, g). Therefore, if p belong to the face where ix + jy + a ij = f (x, y) and ix + jy + b ij = g(x, y), then it follows from (5.10) that the coefficients in monomial ix + jy in G p f, G p g differ by at most ρ(f, g).
Let p belong to different faces in Without loss of generality we may suppose that i = j = 0 and p = (0, 0). Therefore, Other inequalities for the coefficients can be obtained similarly.

A lift of a wave operator G p in characteristic two
Let K be a field with a valuation map val : K * → R. We use the convention val(a+b) ≥ min(val(a)+ val(b)), val(0) = +∞. To each polynomial we associate the tropical polynomial Historically, operators G p appeared as continuous incarnations of waves in sandpiles, see [10]. However, it is naturally to ask about their "detropicalized" version S p , p ∈ (K * ) 2 , namely, how to lift G p to the ring on polynomials (or series) over K.
We managed to do that only in characteristic two. The formula is as follows: if F (p) = 0 and S p F = F if F (p) = 0. We multiply the points coordinatewise.
Theorem 1. For each F ∈ K[x, y] and p ∈ (K * ) 2 the following condition holds Remark 7.1. It is easy to check that (S p F )(p) = 0, which implies that C(Trop(S p F )) passes through p. In turn it implies that S p S p F = S p F .
Proof. Suppose that F (x, y) = A ij X i Y j and p = (p 1 , p 2 ). Suppose that A kl X k Y l is the only monomial with minimal valuation at p (i.e. C(Trop(F )) does not pass through val(p)). Then therefore the valuation of all coefficients for ix+jy of S p f and f are the same except kx+ly. Presenting F (z) near p as F (x, y) = A kl x k x l + G(x, y) we complute the new coefficient for kx + ly as and val(G(p)/F (p)) coincides with the expression for c in Lemma 5.9. Note that if two or more valuations of monomials of F are equal at p, then val(G(p)/F (p)) = 0 and no one coefficient of Trop(F ) changes.
Partial motivation to introduce the operators S p was to prove the finiteness of the dynamic of G pi . Some kind of stabilization (in the smallest terms) for S pi would imply the following finiteness property for G pi .

Contracting a face
By a change of coordinates for a function f : where a, b, c, d, n, m ∈ Z, e, f, k ∈ R, ad − bc = 1. Applying SL(2, Z)-change of coordinates and homothety we may suppose that S is the interval with endpoints (0, 0), (1, 0). We may assume, then, that the neighborhood of S is locally coincide with C(f ), wheref (x, y) = min(0, y, x + n 1 y + c 1 , −x + n 2 y + c 2 ), n 1 , n 2 ∈ Z, c 1 , c 2 ∈ R, because both endpoints of S are smooth vertices of C(f ). Since the endpoints of S are (0, 0), (1, 0), we see that c 1 = 0, c 2 = 1. We suppose that Φ is the face where the function 0 + 0x + 0y is the least monomial inf .
The curve C(Add ct 0,0f ) in the neighborhood of S is given by the tropical polynomial f t (x, y) = min(ct, y, x + n 1 y, −x + n 2 y + 1).
For small t > 0 denote by S t the side of Φ t (recall that Φ t is a face of the curve C(Add ct 0,0f )) which is close and parallel to the side S of the face Φ. It is easy to find the coordinates of the vertices of S t by direct calculation: they are (ct(1 − n 1 ), ct) and(ct(n 2 − 1) + 1, ct). The length of S t is therefore ct(n 2 − 1) + 1 − ct(1 − n 1 ) = 1 + ct(n 1 + n 2 − 2). We just proved the following lemma.
Lemma 8.5. In the above notation, two facts are equivalent: • S t is shorter then S for small t > 0, • n 1 + n 2 < 2.
Corollary 8.6. (Used on pages [11,11,11]) For the above situation there are three cases: a) n 1 + n 2 < 0, this corresponds to collapsing the face Φ to p as t → 1, b) n 1 + n 2 = 0, corresponds to collapsing the face Φ to a (possibly degenerate) interval containing p as t → 1, c) n 1 + n 2 = 1, note that in this case (1, n 1 ) + (−1, n 2 ) = (0, 1). Definition 8.7. We say that a continuous family of tropical curves has a nodal perestroika (see Figure 4) if all the curves, except one, are smooth, and non-smooth curve has only one nodal point, and the family near it is given by min(x, y, t, x + y) for t ∈ [−ε, ε] up to SL(2, Z)-change of coordinates. Proof. The combinatorial type of Φ t can only change when at least one of the sides of the Φ t is getting shrinked to a point for some t. Choose the minimal such t = t 0 , and denote one of the shrinking sides by S. Corollary 8.6 tells us that cases a), b) correspond to collapsing the face, so t 0 = 1, hence in these cases the lemma is proven.
We assume that t 0 < 1 and the case c) in Corollary 8.6 takes place. If neither S 1 nor S 2 gets contracted when we pass from C(f ) to C(Add ct0 ij f ), then we see a nodal perestroika (Definition 8.7). If S 2 is contracted by passing from C(f ) to C(Add ct0 ij f ), then the direct computation using Corollary 8.6 c) implies that the side S 3 of Φ, which is next after S 2 , is parallel to S 2 and therefore the whole face Φ is contracted by Add ct0 ij which is a contradiction. The case when S 1 is contracted is handled by the same argument.

Q-polygons
Definition 9.1. Let ∆ ⊂ R 2 be a finite intersection of half-planes (at least one) with rational slopes. We call ∆ a Q-polygon if it is a closed set with non-empty interior.
Definition 9.2. (Used on pages [12,13,14]) We say that a tropical series f on Ω is presented in the small canonical form if f is written as where all a ij are taken from the canonical form and B f ⊂ A Ω consists of monomials ix + jy + a ij which are equal to f at at least one point in Ω • .
Remark 9.5. (Used on pages ) Note that for a Q-polygon ∆, the small canonical form of the function l ∆ is a ∆-tropical polynomial, i.e. it has only finite number of monomials. It follows from the estimate in Proposition 5.6 that the small canonical form of G P f is a ∆-tropical polynomial too, for all ∆-tropical polynomials f .
Let us fix a Q-polygon ∆. Consider a ∆-tropical polynomial f in the small canonical form (Definition 9.2). Let us analyze the behavior of f near the boundary.
In the neighborhood of each side S of ∆ the function f can be locally written as (x, y) → ix+jy+a ij , where (i, j) ∈ B f and the vector (i, j) is orthogonal to S. This integer vector (i, j) is a multiple of a certain primitive vector, i.e. (i, j) = m f (S)n(S), where n(S) is the inward primitive normal vector to S of ∆ and m f (S) ∈ Z >0 is a number. Thus, we constructed the function m f on the set S(∆) of the sides of ∆, m f : S(∆) → Z >0 . Definition 9.6. (Used on pages [15,18,19]) The aforementioned function m f is called the quasi-degree for the ∆-tropical curve C. Proof. Let {S k } n k=1 be the sides of ∆. Suppose that each side S k is given by i k x + j k y + a k = 0 and all these linear functions are non-negative on ∆. Choose small δ > 0. For each k = 1, . . . , n we consider the following tropical polynomial: The tropical curve defined by f k is the collection of d(S k ) − 1 lines parallel to S k with distance δ between them. Define g as g(x, y) = min ε/2, min k=1,...,n f k (x, y) .
Clearly, C(g) ∩ ∆ is contained in the ε-neighborhood of ∂∆. It is a local calculation near each corner that C(g) ⊂ R 2 is a smooth tropical curve: since the quasi-degree is nice, so near a corner of ∆, C(g) is given locally by where n = d(S k ) δ . Such a curve has an edge locally given by x = ε/2 and, if δ is small enough, d(S k ) − 1 edges locally given by y = k n , 1 ≤ k ≤ d(S k ), and these edges meet in smooth position, see Figure 6 for an illustration.  [13]) If Ω is bounded, then for any ε > 0 the set Ω ε = {x ∈ Ω|f Ω,P ≥ ε} is a Q-polygon and f Ω,P | Ωε is a tropical polynomial. Figure 6. Left: the curve corresponding to the function g from Theorem 2, near a corner, d(S k ) = 4. Each vertex V of the curve is smooth because g is locally presented as min(y, kx, (k + 1)x) near V . Right: an example of C(g) for g in Lemma 12.3. Colored corners symbolize that a quasidegree was not nice, and we made blow-ups at these corners.
Proof. Note that G P 0 Ω = f Ω,P (x) by the definition of the latter, so it follows from Lemma 5.7 that f Ω,P is continuous and vanishes at ∂Ω. Since Ω is bounded, the set f Ω,P = ε is a curve disjoint from ∂Ω. We claim that the intersection of Ω ε with C(f Ω,P ) is a graph with a finite number of vertices. Suppose the contrary. Then a sequence of vertices of this graph converges to a point z ∈ Ω • . Thus, there is no neighborhood of z where the series f Ω,P can be represented by a tropical polynomial, which is a contradiction with Definition 2.2. The finiteness of the number of vertices implies that there is only a finite number of monomials participating in the restriction of f Ω,P to the domain Ω ε , therefore the restriction is a tropical polynomial. Lemma 10.3. (Used on pages ) In the above hypothesis, we extend f Ωε,P to Ω using the presentation of f Ωε,P in the small canonical form (Definition 9.2). In the hypothesis of the previous lemma, if f Ω,P (p) ≥ ε for each p ∈ P , then we have f Ω,P = f Ωε,P + ε on Ω ε . Also f Ωε,P + ε ≥ f Ω,P on Ω.

Proof.
On Ω ε we have that f Ω,P − ε ≥ f Ωε,P by the definition of the latter. Then, two functions f Ωε,P + ε, f Ω,P are equal on ∂Ω ε and by the previous line the quasi-degree of f Ωε,P is at most the quasi-degree of (f Ω,P − ε)| Ωε . Hence f Ω,P can not decrease slowly than f Ωε,P when we move from ∂Ω ε towards ∂Ω. Therefore f Ωε,P + ε ≥ f Ω,P on Ω \ Ω ε . Since f Ωε,P + ε ≥ 0 on Ω we obtain the estimate f Ωε,P + ε ≥ f Ω,P on Ω which concludes the proof.
Note that a Q-polygon is not necessary compact. It is easy to verify that a Q-polygon is admissible (Definition 2.8). The next lemma provides us with a family of compact Q-polygons exhausting Ω.
Lemma 10.4. (Used on pages [14]) For any compact set K ⊂ Ω • such that P ⊂ K and for any ε > 0 small enough there exists a Q-polygon Ω ε,K ⊂ Ω such that B 3ε (K) ⊂ Ω ε,K and the following holds: Proof. Note that if Ω ⊂ Ω, then f Ω ,P ≤ f Ω,P automatically. We list several possible cases. A) Ω is a compact set, see Lemma 10.2. If Ω is not compact, then it is possible that B) Ω is a half-plane with the boundary of rational slope. Otherwise, ∂Ω has two asymptotes: C) of rational slope, D) of irrational slope, E) one of asymptotes is of rational slope and another is not.
Let M = max K f Ω,P . It follows from Lemma 3.4 that the set I of monomials (i, j) ∈ Z 2 such that there exists a ij such that (a ij + ix + jy)| K ≥ 0 and a ij + ix + jy < M at a point of K is finite. Only these monomials may contribute to f Ω,P | K and we are going to study their coefficients.
For each (i, j) ∈ I there are three possibilities: i) for some c the line {c+ix+jy = 0} is an asymptote of ∂Ω; ii) for a compact set K , containing P and big enough, ix + jy − min K ∩Ω (ix + jy) ≥ 0 on Ω; iii) for a compact set K , containing P and big enough, ix + jy − min K ∩Ω (ix + jy) > M on K.
In the case B) this implies that for K big enough the only monomials which contribute to f K ∩Ω are the multiples of the monomial giving ∂Ω, therefore f Ω,P = f Ω∩K ,P , which reduces the proof to A).
The case D) is handled similarly: ii) is not possible, but i) implies that f Ω,P ≤ f Ω∩K ,P on K, which proves that f Ω,P = f Ω∩K ,P on K.
In the case C) we prove that {f Ω,P ≥ ε} is a Q-polygon for ε > 0 small enough. Indeed, let one of the asymptotes is given by L = {kx + ly + a kl = 0}. Then, for the points z ∈ L far enough from P we have that the distance between z and ∂Ω is at least ε/2 and therefore by Lemma 3.4 the set I of monomials a ij + ix + jy which are non-negative on Ω and less than ε at z is finite. Therefore, by taking K big enough and containing all the points of intersection of the support lines to ∂Ω with directions in I (if there is no a point of intersection, it means that this is another asymptote and this is handled easily), we may assume that f Ω,P = ε is given by L far enough from P and the same for another asymptote. Therefore the curve {f Ω = ε} is a Q-polygon.
The last case, E) is handled as follows: we take K as above and then find a line L(x, y) = 0 with a rational slope close to the irrational slope of an asymptote, such that L| K ∩Ω ≥ 0 and we reduce the case to D) by considering Ω ∩ {L ≥ 0} instead of Ω.
Corollary 10.5. Lemma 10.4 implies that for ε > 0 small enough the tropical curves defined by f Ω,P and f Ω ε,K ,P coincide on K, i.e.
We say that this blow-up is made with respect to the lattice point (i, j). Note that ∆ ⊂ ∆. We say that ∂∆ \ ∂∆ (i.e. the new side of ∆ obtained as cutting the corner at O) is the side, dual to the vector (i, j).
Note that we do not require that (i, j) is a primitive vector. This will be important in Lemma 12.3.
Remark 11.4. Note that if Λ is unimodular (Definition 8.3) then p 1 q 2 − p 2 q 1 = 1 and there exists a preferred direction (p 1 + p 2 , q 1 + q 2 ) to perform a blow-up which produces two unimodular corners near the vertex of Λ.
Let f be any Λ-tropical polynomial written in the small canonical form (Definition 9.2). So, supp(f ) ⊂ A Λ and is finite. Recall that O = (0, 0) is the corner of Λ.
Lemma 11.5. (Used on pages [15,16]) Consider any ε > 0 small enough. There exist δ > 0, Proof. We consider the case (p 1 , q 1 ) = (1, 0), (p 2 , q 2 ) = (0, 1), the general case can be handled in the same way. If ε is small enough, then we have where A ⊂ A Λ . It is enough to prove the statement for (p, q) = (1, N ), i.e. that if N is big enough and δ > 0 is small enough, then The cone Λ is dissected on regions where each of p i x + q i y is the minimal monomial. All these sectors except one satisfy y > cx for a constant c depending on p i , q i . Therefore if N is big enough then x + N y > (p i + 1)x + (q i + 1)y > p i x + q i y + δ if x or y is bigger than δ. The only region where we do not have the estimate y > cx is the region where the minimal monomial p i x + q i y satisfies p i = 0. In this region, again, x + N y > q i y + δ if x or y is bigger than δ and N is big enough.

Nice tropical series
Definition 12.1. (Used on pages [15,16,16]) Let f be a ∆-tropical series. We say that f is nice if all the corners of ∆ are unimodular (Definition 8.3) and the quasi-degree (Definition 9.6) m f is nice (Definition 9.8).
Lemma 12.2. (Used on pages [16]) Let ∆ be a Q-polygon. Suppose that f is a nice ∆-tropical series. Then, C(f ) has exactly one edge of weight one passing through each corner of ∆.
Proof. Consider any ordering {(i k , j k )} ∞ k=1 of primitive vectors in A Λ \{(p 1 , q 1 ), (p 2 , q 2 )} such that i 2 k+1 + j 2 k+1 ≥ i 2 k + j 2 k for any pair of consecutive (with respect to this order) primitive vectors. Choose δ > 0 small enough and denote by Λ k the δ-blow-up of Λ k−1 with respect to the vector n k (i k , j k ) where n k ∈ N is chosen in such a way that ||n k (i k , j k )|| ≥ N (see Lemma 11.5).
Note that Λ k−1 contains k corners but only one of them can be blow-upped using the direction (i k , j k ); so there is no ambiguity.
We construct the following sequence {f k : Λ k → R} ∞ k=1 of functions. The function f 0 is taken to be f on Λ 0 = Λ. We take f k to be Because of the choice of n k we know that f k and f k−1 are equal outside of a small neighborhood of O. The number n k represents the quasi-degree of f n , n > k on the side dual to the vector n k (i k , j k ). By Lemma 11.5 for large k all this n k can be chosen to be 1. Therefore from the construction it is clear that f n is nice on Λ n for some n big enough.
Proposition 12.4. Let ∆ be a Q-polygon. Consider a sequence of operators G q1 , G q2 , . . . , G qm where q 1 , q 2 , . . . , q m are (not necessary distinct) points in ∆ • . We will use the following notation Then, for each ε > 0 small enough there exists a unimodular Q-polygon ∆ ⊂ ∆ such that • G0 ∆ is nice (Definition 12.1) on ∆ , Proof. Consider f = G0 ∆ . Using Lemma 12.3, we make necessary blow-ups at each corner of ∆, constructing in this way a Q-polygon ∆ ⊂ ∆ and a nice functionf on ∆ . By construction, f =f near P . Therefore G0 ∆ ≤f and hence G0 ∆ is nice on ∆ . Clearly G0 ∆ ≥ G0 ∆ and we might do blow-ups in so small neighborhood of the corners of ∆ such that G0 ∆ < ε on ∂∆ which implies the third assessment. The second assessment follows from Lemma 6.6, because ρ(0 ∆ , 0 ∆ ) is arbitrary small for small ε, if these function are written in the canonical form.
13. Coarse smooth approximation of the dynamic G P Let ∆ be a Q-polygon and g be a nice (Definition 12.1) ∆-tropical series, such that C(g) is a smooth tropical curve. Let q i ∈ ∆ • , i = 1, . . . m, and f = G qm G qm−1 . . . G q1 g. Since each G q k is the application of Add e k i k ,j k for some e k > 0, we can write Proof. Since m f = m g , we do not apply operators G qi in the regions adjacent to the boundary of ∆. The only two possibilities how the tropical curve can become non-smooth during our procedure in (15.1) is appearance of a non-smooth vertex inside ∆ • and appearance of an edge with weight bigger than one inside ∆ • or at the corners of ∆.
To satisfy ε-closeness, it is enough that mM h < ε. It follows from Lemma 8.8 that a non-smooth vertex or an edge with weight bigger than one in ∆ • can appear only by contracting a face. We can decrease the constants e i in (15.1) by any small positive numbers, such that no G • q k contracts a face, this eliminates a part of the problems with smoothness inside ∆ • . To be sure that this decreasing did not change the incidence between faces and points q i in the process it is enough to choose M such that mM h (the total change of function) would be less than the minimal non-zero distance between one of the points q 1 , . . . , q m and the tropical curves G qi . . . G q1 g, i = 1, . . . , m. Finally, f m is nice on ∆ and, by Lemma 12.2, the tropical curve C(f m ) has no edges of weight bigger than one at the corners of ∆.

Tropical symplectic area
One may ask what are intrinsic properties of f Ω,P . We will prove that the curve C(f Ω,P ) solves a sort of Steiner problem, see Corollary 14.7.
Definition 14.1 (See [24]). The tropical symplectic area of an interval L ⊂ R 2 with a rational slope is Area(L) = ||L|| · ||v||, where || − || denotes a Euclidean length and v is a primitive integer vector parallel to l. If C is an Ω-tropical curve, then its tropical symplectic area is the weighted sum of areas for its edges e, i.e.
where m e is the weight of the edge e (Definition 8.1). This area may be infinite as well, if C contains infinite number of edges and the series diverges or C has edges of infinite length.
The motivation for this definition is as follows. Recall that an amoeba of an algebraic curve S in the algebraic torus (C * ) 2 is an image of S in R 2 under the logarithm map log t (z 1 , z 2 ) = (log t |z 1 |, log t |z 2 |). Consider a family {S t } of algebraic curves in (C * ) 2 for t > 0. We say that {S t } tropicalizes to the tropical curve C if the family log t S t ⊂ R 2 converges to C when t → ∞. It could seem that the tropicalization of {S t } is defined only as a set. In fact, the multiplicities for the edges of C can be also canonically restored from the family S t .
Consider the following symplectic form on (C * ) 2 : This justifies the name " tropical symplectic area": it is the main part in the asymptotic for symplectic areas.
Proof. For a large t, log t (S t ) is in a small neigborhood of the tropical curve C. Moreover, S t itself will be close to a certain lift of C to the torus (C * ) 2 . It is performed by lifting each edge with a slope (p, q) to a piece of holomorphic cylinder {(z p , z q )|z ∈ C} translated by the action of the torus. This lift is called a complex tropical curve (see [19] for the details).
Therefore, we can compute the area of S t near the limit by looking at the areas of the cylinders. There also can be minor corrections coming from the vertices of C but the corrections are small with respect to log t and so do not appear in the final statement.
To complete the proof we need to compute the contribution from each edge in C ∩ B. It is clear that for each segment in C ∩ B the area of its lift is proportional to the length of the segment. So if we show that the area of the lift for the interval going from the origin to the integer vector (p, q) is equal to 4π 2 (p 2 + q 2 ) log t then we will be done. This computation is given by application of the following lemma for both parts of ω.  Then the left hand side of the equality we are proving is equal to Remark 14.4. The specific choice for ω is not crucial while it is invariant under the action of (C * ) 2 . Indeed, if ω is an arbitrary 2-form then its restriction to any holomorphic curve will not have contributions from pure holomorphic and anti-holomorphic parts of ω . So we can think that ω is a (1, 1)-form.
There is a two dimensional family of torus-invariant (1, 1)-forms. Different choices for ω from this family correspond to coordinate dilatations on the level of tropical curves.
Proposition 14.2 suggests us that symplectic area for tropical curves should be deformation invariant. Indeed, this should follow from the fact that the 2-form ω is closed. And indeed, we can prove the deformation invariance directly. Proof. Any deformation C s locally can be decomposed into the elementary ones. Near each vertex of C, an elementary deformation is a process of moving and shortening two edges while growing the one in the opposite direction (see Figure 8).
Globally this corresponds to enlarging a coefficient for a tropical polynomial. For example on Figure  4 we change the coefficient for the central region.
Up to a scaling, an elementary deformation simply replaces the union of segments [0, v 1 ] an [0, v 2 ] by a single segment [0, v 1 + v 2 ]. Here v 1 and v 2 are the primitive (or appropriate multiples of primitive) vectors for the edges we are moving. Denote by w i the projection of v 1 + v 2 on the line spanned by v i (see Figure 9). Then after the deformation the two edges together loose of their tropical symplectic area. On the other hand, the growing edge contributes exactly |v 1 + v 2 | 2 to the symplectic area of the deformed curve.
Let us get back to our specific case. Let ∆ be a compact Q-polygon and f be a ∆-tropical polynomial with quasidegree m f (Definition 9.6). Then we can deform C(f ) to the union of all edges e of the polygon taken with the multiplicities m f (e). This observation together with the deformation-invariance (Lemma 14.5) proves the following lemma. Corollary 14.7 (cf. Theorem 3 in [11]). If ∆ is a compact Q-polygon and P ⊂ ∆ • is a finite collection of points, then the tropical curve C(f ∆,P ) has the minimal tropical symplectic area among all ∆-tropical curves passing through the configuration of points P . Indeed, the tropical symplectic area is determined by the quasidegree, and f Ω,P has the minimal on each side of ∆ degree among the ∆-tropical series non-smooth at P .
We should mention that the tropical symplectic area of a tropical curve already appeared in physics under the name of mass of a web [14], where a web is a direct analog of a tropical curve. Only for curiosity we present a part of the dictionary between tropical objects and the field theory. We quote [2]: "On the other hand, we already know that shrinking an internal face of a string web corresponds to a bosonic zero mode ... so the mass of the web is independent of this deformation." -this reminds us the operator G p , shrinking a face. This mass is also presented as a trace of some operator (BPS-formula, ibidem), and the area of a face in ∆ \ C(f ) is interpreted there as the "tension of a monopolic string".

Summary
For easy reference we formulate here a theorem, which summarizes most things about coarsening that we need in [10].
Theorem 3. Choose ε > 0. For a given Q-polygon ∆ and a finite set P ⊂ ∆ • there exist a Q-polygon ∆ ⊂ ∆ and a ∆ -tropical polynomial g such that a) g| ∆ < ε, the curve C(g) is smooth, and G P g = G P 0 ∆ , b) G P g is ε-close to G P 0 ∆ . Using Proposition 6.1 let us write G = G qm G qm−1 . . . G q1 g such that G is ε-close to G P g and their quasi-degrees (Definition 9.6) coincide. c) Then, during the calculation of G we never apply a wave operator for a face which has a common side with ∂∆ . Note that in the product G qm G qm−1 . . . G q1 g each G q k is the application of Add e k i k ,j k for some e k > 0, i.e. we increase the coefficient in the monomial i k x + j k y by e k . So we have (Used on pages [16,16,16,19]) G qm G qm−1 . . . G q1 g = Add em imjm Add em−1 im−1jm−1 . . . Add e1 i1j1 g. Denote f 0 = g, f k+1 = Add e k −M h i k j k f k = G • q k (f k ). d) Then there exists a constant M such that for any h > 0 small enough all the tropical curves defined by f k , k = 1, . . . , m are smooth or nodal (Definition 8.4) on ∆ as well as each tropical curve in the family during the application of G • q k to f k (Remark 5.12); and e) the tropical curve defined by f m is ε-close to the tropical curve defined by G qm G qm−1 . . . G q1 g.
Theorem 2 gives a), b). Then, c) follows from the fact that the quasi-degrees of G, G P g coincide. The content of Proposition 13.2 is d),e).