Triangulations of polytopes and algebraic geometry

ABSTRACT

The interaction between polyhedral combinatorics and algebraic geometry is a classical theme. Two examples from the 70's, in which each discipline has benefited from the other, are the Bernstein-Kouchnirenko-Khovanski Theorem on the number of roots of a generic sparse system via mixed volumes and the algebraic proofs by Stanley of the Upper Bound Theorem for simplicial spheres and the g-theorem for simplicial polytopes.

REFERENCES

• 1: M. Develin, F. Santos, and B. Sturmfels, On the tropical rank of a matrix, preprint December 2003, 20 pages, arxiv.org/math.CO/0312114.
• 2: M. Develin, and B. Sturmfels, Tropical convexity, preprint August 2003, 19 pages, arxiv.org/math.CO/0308254.
• 3: F. Santos, Non-connected toric Hilbert schemes, preprint March 2002. to appear in Mathematische Annalen.
• 4: F. Santos, The Cayley Trick and triangulations of products of simplices, preprint December 2003. To appear in Proceedings of the Joint Summer Research Conference on Integer Points in Polyhedra. Geometry, Number Theory, Algebra, and Optimization, edited by A. Barvinok, M. Beck, C. Haase, B. Reznick, and V. Welker, Contemporary Mathematics, American Mathematical Society.
• 5: B. Sturmfels, Gröbner bases and convex polytopes, University Series Lectures 8, American Mathematical
• 6: O. Viro, Real algebraic plane curves: Constructions with controlled topology, Algebra i Analiz, 1 (1989), 1--73. Translation in Leningrad Math.J., 1 (1990), pp.1059--1134.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.1 Combinatorics
          Subjects: Combinatorial algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

General Terms:
Theory

Keywords:
toric Hilbert scheme, tropical geometry