The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise   polynomial
  We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory. 
Federico Ardila and Erwan Brugalle
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