Counting tropical curves in $\mathbb{P}^1\times\mathbb{P}^1$: computation & polynomiality properties

Abstract: Counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$ with fixed contact order with the toric boundary and satisfying point conditions can be determined with tropical methods by Mikhalkin. If we require that our curves intersect the zero- and infinity-section only in points of contact order $1$, but allow arbitrary contact order for the zero- and infinity-fiber, the corresponding numbers reveal beautiful structural properties such as piecewise polynomiality, similar to the case of double Hurwitz numbers counting covers of $\mathbb{P}^1$ with special ramification profiles over zero and infinity by Ardila and Brugall\'e. This result was obtained using the floor diagram method to count tropical curves. Here, we expand the tropical tools to determine counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$. We provide a computational tool (building on Polymake by Gawrilow and Joswig) that determines such numbers of tropical curves for any genus and any contact orders via a straightforward generalization of Mikhalkin's lattice path algorithm. The tool can also be used for other toric surfaces. To enable efficient computations also by hand, we introduce a new counting tool (for the case of rational curves with transverse contacts with the infinity section) which can be seen as a combination of the floor diagram and the lattice path approach: subfloor diagrams. We use both our computational tool and the subfloor diagrams for experiments revealing structural properties of these counts. We obtain first results on the (piecewise) polynomial structure of counts of rational curves in $\mathbb{P}^1\times\mathbb{P}^1$ with arbitrary contact orders on the zero- and infinity-fiber and restricted choices for the contact orders on the zero- and infinity-section.