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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Enumerative tropical algebraic geometry in $\mathbb {R}^2$
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by Grigory Mikhalkin
J. Amer. Math. Soc. 18 (2005), 313-377
Published electronically: January 20, 2005


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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Bibliographic Information
  • Grigory Mikhalkin
  • Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, M5S 3G3 Canada and St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011 Russia
  • Address at time of publication: IHES, Le Bois-Marie, 35, route de Chartres, Bures-sur-Yvette, 91440, France
  • Email:
  • Received by editor(s): January 31, 2004
  • Published electronically: January 20, 2005
  • Additional Notes: The author would like to acknowledge partial support of the NSF and NSERC
  • © Copyright 2005 Grigory Mikhalkin
  • Journal: J. Amer. Math. Soc. 18 (2005), 313-377
  • MSC (2000): Primary 14N35, 52B20; Secondary 14N10, 14P25, 51M20
  • DOI:
  • MathSciNet review: 2137980