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We define the double Gromov-Witten invariants of Hirzebruch surfaces inanalogy with double Hurwitz numbers, and we prove that they satisfy a piecewisepolynomiality property analogous to their 1-dimensional counterpart.Furthermore we show that each polynomial piece is either even or odd, and wecompute its degree. Our methods combine floor diagrams and Ehrhart theory.

The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial