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Tropical Algebraic Geometry

Yaghmayi, Keyvan | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41546 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Rastegar, Arash
  7. Abstract:
  8. Tropical algebraic geometry is a fairly new branch in geometry which is called so in honor of Brazilian mathematician Imre Simon who was pioneer of this field.The set equipped with the addition and multiplication is a semifield.Algebraic geometry objects like algebraic varieties can be defined over.Polynomials and rational functions are defined over.The functions that they define are piecewise linear and concave functions and the set of points where they are nonlinear is a tropical variety which is a concave polyhedral. Thus, tropical algebraic geometry is a piecewise linear version of algebraic geometry.Another approach to tropical algebraic geometry comes back to the works of Russian mathematicians such as V.P. Maslov, M.M. Kapranov and G.L. Litvinov. In the set defined above, we can replace with and change min to max in the tropical sum. Then is the image of the map when . This deformation is known as Maslov dequantization of non-negative real numbers. In this approach tropical algebraic geometry is dequantization of algebraic geometry.In this thesis, we introduce some tropical objects and structures like tropical polynomials, tropical varieties, tropical Picard group, etc. and give proof to some analogous well-known theorems of algebraic geometry in tropical geometry e.g. the fundamental theorem of tropical algebra, the tropical Riemann-Roch theorem, the Bezout’s theorem, etc. We also show some applications of tropical geometry in enumerative geometry like Gromov-Witten invariants and Hilbert’s sixteenth problem.

  9. Keywords:
  10. Tropical Algebraic Geometry ; Tropical Varieties ; Polyhedral Complexes ; Tropical Basis ; Amoebas ; Algebraic Varieties Dequantization ; Idempotent Rings

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