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...Kontsevich s formula and the wdvv equations in tropical geometry andreasgathmannandhannahmarkwig abstract using gromov witten theory numbers of complex plane ra tional curves degree d through general given points can be computed recursively with that follows from so called this paper we establish same results entirely language particular shows how concepts moduli spaces stable maps evaluation forgetful morphisms inter section multiplicities their invariance under deformations carried over to world introduction for let n number rational projective p pass position about years ago has shown these are by initial value andtheequation x see claim main tool deriving is i e associativity quantum cohomol ogy stated modern terms idea as parametrized m whose bijection tuples c f where dis tinct smooth on a nodal curve morphism stability condition if there map sends stabi lization important point now space pointed simply line therefore two linearly equivalent divisors hence inverse images divisor ...