On the piecewise quasipolynomiality of double tropical Welschinger invariants

TL;DR

This paper proves that double tropical Welschinger invariants of certain toric surfaces are piecewise quasipolynomial and introduces new combinatorial invariants with similar properties.

Contribution

It confirms a conjecture for all h-transverse polygons and defines new combinatorial Welschinger-type numbers with piecewise quasipolynomiality.

Findings

Proved the conjecture for all toric surfaces with h-transverse polygons.

Established piecewise quasipolynomiality of new combinatorial Welschinger-type numbers.

Extended the understanding of tropical Welschinger invariants in algebraic geometry.

Abstract

Ardila and Brugall\'e 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.