Enumerative tropical algebraic geometry in ℝ²

Mikhalkin, Grigory

doi:10.1090/S0894-0347-05-00477-7

Enumerative tropical algebraic geometry in $\mathbb {R}^2$
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by Grigory Mikhalkin

J. Amer. Math. Soc. 18 (2005), 313-377

DOI: https://doi.org/10.1090/S0894-0347-05-00477-7

Published electronically: January 20, 2005

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Abstract:

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space $\mathbb {R}^n$ and holomorphic curves with certain piecewise-linear graphs there.

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