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Recently, Goettsche and Shende defined a notion of "refined Severi degree", replacing the degree of the Severi variety by a univariate polynomial. We give a (conjectural) interpretation of these invariants for plane curves in terms of tropical geometry: Goettsche and Shende's invariant is computed by enumeration of tropical curves, weighted by a new (Laurent) polynomial multiplicity. This weight specializes to Mikhalkin's 'complex' and 'real' multiplicity of tropical curves.
This is joint work with Lothar Goettsche.
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