Polynomiality properties of tropical refined 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 Göttsche, further extended Göttsche Schroeter in the case rational curves. In this paper, we study polynomial behavior coefficients these invariants. We prove that small codegree are polynomials Newton polygon curves under enumeration, when one fixes genus latter. This provides surprising reappearance, dual setting, so-called node conjecture. Our methods, based on floor diagrams Mikhalkin first author, entirely combinatorial. Although combinatorial treatment needed here is different, follow overall strategy designed Fomin developed Ardila Block. Hence our results may suggest phenomena geometry have not been studied yet. particular curves, extend polynomiality including extra parameter \(s\) recording number \(\psi\) classes. Contrary to with respect \( \Delta\), be expected from considerations Welschinger pleads favor geometric definition Göttsche-Schroeter invariants.Mathematics Subject Classifications: Primary 14T15, 14T90, 05A15; Secondary 14N10, 52B20Keywords: invariants, geometry, Gromov-Witten