Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and Göttsche, and further extended by Göttsche and Schroeter in the case of rational curves.In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a somehow surprising resurgence, in some sort of dual setting, of the so-called node polynomials and Göttsche conjecture. Our methods are entirely combinatorial, hence our results may suggest phenomenons in complex enumerative geometry that have not been studied yet.In the particular case of rational curves, we extend our polynomiality results by including the extra parameter s recording the number of ψ classes. Contrary to the polynomiality with respect to ∆, the one with respect to s may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of Göttsche-Schroeter invariants.

Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and Göttsche, and further extended by Göttsche and Schroeter in the case of rational curves.In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a surprising reappearance, in a dual setting, of the so-called node polynomials and the Göttsche conjecture. Our methods, based on floor diagrams introduced by Mikhalkin and the first author, are entirely combinatorial. Although the combinatorial treatment needed here is different, we follow the overall strategy designed by Fomin and Mikhalkin and further developed by Ardila and Block. Hence our results may suggest phenomena in complex enumerative geometry that have not been studied yet.In the particular case of rational curves, we extend our polynomiality results by including the extra parameter s recording the number of ψ classes. Contrary to the polynomiality with respect to ∆, the one with respect to s may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of Göttsche-Schroeter invariants.

In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in RP 2 . In order to study the rigid isotopy classes of nodal rational curves of degree 5 in RP 2 , we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface Σ 3 and the corresponding nodal real dessin on CP 1 /(z → z). The dessins are real versions, proposed by S. Orevkov [10], of Grothendieck's dessins d'enfants. The dessins are graphs embedded in a topological surface and endowed with a certain additional structure.We study the combinatorial properties and decompositions of dessins corresponding to real nodal trigonal curves C ⊂ Σn in real Hirzebruch surfaces Σn. Nodal dessins in the disk can be decomposed in blocks corresponding to cubic dessins in the disk D 2 , which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of real rational quintics in RP 2 . Contents 42 3.1. Hirzebruch surfaces 42 3.2. Positive and negative inner nodal ×-vertices 43 4. Generic quintic rational curves in RP 2 47 4.1. Maximally perturbable curves 50 4.2. (M − 2)-perturbable curves 55 4.3. (M − 4)-perturbable curves 63 4.4. (M − 6)-perturbable curves 65 References 66

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), of Grothendieck’s dessins d’enfants. This classification contains 20 classes determined by the number of ovals of A, the parity of the oval containing the marked point p, the number of ovals that the tangent line $$T_p A$$ T p A intersects, the nature of connected components of $$A\setminus T_p A$$ A \ T p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of $$A\setminus T_p A$$ A \ T p A . We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary can be decomposed in blocks corresponding to cubic dessins in the disk $${\mathbf {D}}^2$$ D 2 , which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of generic pointed quartic curves in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 . This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.