On the piecewise quasipolynomiality of double tropical Welschinger invariants

Abstract

Ardila and Brugallé conjectured that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial. In this work, we prove the conjecture holds in full generality, i.e. for toric surfaces corresponding to h-transverse polygons. Furthermore, we define new combinatorial Welschinger-type numbers for h-transverse polygons and show that they are likewise piecewise quasipolynomial.