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Amoebas of Algebraic Varieties and Tropical Geometry

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Different Faces of Geometry

Part of the book series: International Mathematical Series ((IMAT,volume 3))

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References

  1. O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, http://arxiv.org hep-th/9704170.

  2. O. Aharony, A. Hanany, and B, Kol, Webs of (p, q) 5-branes, five dimensional field theories and grid diagrams, http://arxiv.org hep-th/9710116.

  3. V. I. Arnold, The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms [in Russian], Funk. Anal. Pril. 5 (1971) no. 3, 1–9.

    MATH  Google Scholar 

  4. M. F. Atiyah, Angular momentum, convex polyhedra and algebraic geometry, Proc. Edinburgh Math. Soc. 26 (1983) 121–133.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. M. Bergman, The logarithmic limit set of an algebraic variety, Trans. Am. Math. Soc. 157 (1971), 459–469.

    MATH  MathSciNet  Google Scholar 

  6. D. Bernstein, The number of roots of a system of equations, Funct. Anal. Appl. 9 (1975), 183–185.

    MATH  Google Scholar 

  7. F. Bourgeois, A Morse-Bott Approach to Contact Homology, Dissertation, Stanford University, 2002.

    Google Scholar 

  8. L. Brusotti, Curve generatrici è curve aggregate nella costruzione di curve piane d’ordine assegnato dotate del massimo numero di circuiti, Rend. Circ. Mat. Palermo 42 (1917), 138–144.

    MATH  Google Scholar 

  9. Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, GAFA 2000, Special Volume, Part II, 560–673.

    Google Scholar 

  10. M. Forsberg, M. Passare, and A. Tsikh, Laurent determinants and arangements of hyperplane amoebas, Adv. Math. 151 (2000), 45–70.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994.

    Google Scholar 

  12. A. Harnack, Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), 189–199.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Henriques, An Analogue of Convexity for Complements of Amoebas of Varieties of Higher Codimensions, Preprint, Berkeley, May 2001.

    Google Scholar 

  14. D. Hilbert, Mathematische Probleme, Arch. Math. Phys. (3) 1 (1901), 213–237.

    MATH  Google Scholar 

  15. I. Itenberg, V. Kharlamov, E. Shustin, Welschinger invariant and enumeration of real plane rational curves, http://arxiv.org/abs/math.AG/0303378.

  16. I. Itenberg and O. Viro, Patchworking real algebraic curves disproves the Rags-dale conjecture, Math. Intel. 18 (1996), 19–28.

    MathSciNet  MATH  Google Scholar 

  17. M. M. Kapranov, A characterization of A-discriminantal hypenurfaces in terms of the logarithmic Gauss map, Math. Ann. 290 (1991), 277–285

    MATH  MathSciNet  Google Scholar 

  18. M. M. Kapranov, Amoebas over Non-Archimedian Fields, Preprint, 2000.

    Google Scholar 

  19. R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, http://arxiv.org math.AG/03 11062.

  20. R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae, http://arxiv.org math-ph/0311005.

  21. V. M. Kharlamov, On the classification of nonsingular surfaces of degree 4 in RP 3 with respect to rigid isotopies [in Russian], Funkt. Anal. Pril. 18 (1984), no. 1, 49–56.

    MATH  MathSciNet  Google Scholar 

  22. A. G. Khovanskii, Newton polyhedra and toric varieties, [in Russian], Funkt. Anal. Pril. 11 (1977), no. 4, 56–64.

    MATH  Google Scholar 

  23. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibra-tions, Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publishing, River Edge, NJ, 2001.

    Google Scholar 

  25. V. P. Maslov, New superposition principle for optimization problems, in Semi-naire sur les Equations auc Dérivées Partielles 1985/6, Centre Mathématiques de l’École Polytechnique, Palaiseau, 1986, exposé 24.

    Google Scholar 

  26. G. Mikhalkin, Real algebraic curves, moment map and amoebat, Ann. Math. 151 (2000), 309–326.

    Article  MATH  MathSciNet  Google Scholar 

  27. G. Mikhalkin and H. Rullgård, Amoebas of maximal area, Intern. Math. Res. Notices 9 (2001), 441–451.

    Google Scholar 

  28. G. Mikhalkin, Amoebas of algebraic varieties, a report for the Real Algebraic and Analytic Geometry congress, June 2001, Rennes, France, http://arxiv.org math.AG/0108225.

  29. G. Mikhalkin, Decomposition into Pairs-of-Pants for Complex Algebraic Hyper-surfaces, http://arxiv.org math.GT/0205011 Preprint 2002 [To appear in Topology.]

  30. G. Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634.

    MATH  MathSciNet  Google Scholar 

  31. G. Mikhalkin, Enumerative tropical geometry in R 2, http://arxiv.org math.AG/0312530.

  32. G. Mikhalkin, Maximal real algebraic hypersurfaces (in preparation)

    Google Scholar 

  33. M. Passare and H. Rullgård, Amoebas, Monge-Ampère Measures, and Triangu-lations of the Newton Poly tope. Preprint, Stockholm University, 2000.

    Google Scholar 

  34. I. G. Petrovsky, On the topology of real plane algebraic curves, Ann. Math. 39 (1938), 187–209.

    Google Scholar 

  35. J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50–69, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge, 1998.

    Google Scholar 

  36. J. Richter-Gebert, B. Sturmfels, Th. Theobald, First steps in tropical geometry, http://arxiv.org math.AG/0306366.

  37. V. A. Rohlin, Congruences modulo 16 in Hilbert’t sixteenth problem, [in Russian], Funkt. Anal. Pril. 6 (1972), no. 4, 58–64.

    MATH  MathSciNet  Google Scholar 

  38. L. Ronkin, On zeroes of almost periodic functions generated by holomorphic functions in a multicircular domain [To appear in “Complex Analysis in Modern Mathematics,” Fazis, Moscow, 2000, 243–256].

    Google Scholar 

  39. H. Rullgård, Stratification des espaces de polynômes de Laurent et la structure de leurs amibes, C. R. Acad. Sci. Paris, Série I, 331 (2000), 355–358.

    MATH  Google Scholar 

  40. H. Rullgård, Polynomial Amoebas and Convexity, Preprint, Stockholm Univerity, 2001.

    Google Scholar 

  41. D. Speyer, B. Sturmfels, The tropical Grassmanian, http://arxiv.org math.AG/0304218.

  42. O. Ya. Viro, Real plane algebraic curves: constructions with controlled topology, Leningrad Math. J. 1 (1990), no. 5, 1059–1134.

    MATH  MathSciNet  Google Scholar 

  43. O. Ya. Viro, Dequantization of Real Algebraic Geometry on a Logarithmic Paper, Proceedings of the European Congress of Mathematicians (2000).

    Google Scholar 

  44. G. Wilson, Hilbert’t sixteenth problem, Topology 17 (1978), 53–73.

    Article  MATH  ISI  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Utah, Utah, USA

    Grigory Mikhalkin

  2. University of Toronto, Toronto, Canada

    Grigory Mikhalkin

  3. POMI RAS, St. Petersburg, Russia

    Grigory Mikhalkin

Authors
  1. Grigory Mikhalkin
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Editors and Affiliations

  1. Imperial College of London, London, UK

    Simon Donaldson

  2. Stanford University, Stanford, California

    Yakov Eliashberg

  3. Institut des Haute Etudes Scientifiques, Bures-sur-Yvette, France

    Mikhael Gromov

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Mikhalkin, G. (2004). Amoebas of Algebraic Varieties and Tropical Geometry. In: Donaldson, S., Eliashberg, Y., Gromov, M. (eds) Different Faces of Geometry. International Mathematical Series, vol 3. Springer, Boston, MA. https://doi.org/10.1007/0-306-48658-X_6

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  • DOI: https://doi.org/10.1007/0-306-48658-X_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-306-48657-9

  • Online ISBN: 978-0-306-48658-6

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