  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 G\"ottsche, and further
extended by G\"ottsche 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 G\"ottsche
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 $\psi$
classes. Contrary to the polynomiality with respect to $ \Delta$, 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 G\"ottsche-Schroeter invariants.
