On polynomially integrable planar outer billiards and curves with symmetry property

We show that every polynomially integrable planar outer convex billiard is elliptic.Comment: To appear in Mathematische Annalen. 26 pages. Minor improvement of presentatio

Tropical vertex and real enumerative geometry

We observe that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.Comment: 18 pages, 1 figure; references [3] added, reference [13] update

Refined descendant invariants of toric surfaces

We construct refined tropical enumerative genus zero invariants of toric surfaces that specialize to the tropical descendant genus zero invariants introduced by Markwig and Rau when the quantum parameter tends to $1$. In the case of trivalent tropical curves our invariants turn to be the Goettsche-Schroeter refined broccoli invariants. We show that this is the only possible refinement of the Markwig-Rau descendant invariants that generalizes the Goettsche-Schroeter refined broccoli invariants. We discuss also the computational aspect (a lattice path algorithm) and exhibit some examples.Comment: 30 pages, 7 figures; matches the published versio