Enumerative Real Algebraic Geometry

Frank Sottile   sottile@tamu.edu
4 August 2002

2000 Mathematics Subject Classification 14P99, 12D10, 14N10, 14N15, 14M15, 14M25, 14M17

Table of Contents

Summary
  1. Introduction
  2. Polynomial Systems
    1. Sparse Polynomial Systems
    2. The Polyhedral Homotopy Algorithm
    3. Real Solutions to Sparse Polynomial Systems
  3. Enumerative Real Algebraic Geometry
    1. Enumerative Real Algebraic Geometry
    2. Problems not involving general conditions
    3. The Stewart-Gough Platform
    4. Real rational cubics through 8 points in R2
    5. Common tangent lines to Spheres in Rn
  4. Schubert Calculus
    1. The Schubert Calculus of Lines
    2. The Special Schubert Calculus
    3. Further Extensions of the Schubert Calculus
  5. The Conjecture of Shapiro and Shapiro
    1. The Conjecture of Shapiro and Shapiro for Grassmannians
    2. Rational functions with real critical points
    3. Extensions of the Conjecture of Shapiro
  6. Lower Bounds in the Schubert calculus
    1. Real Degree of Grassmann varieties
    2. Lower bounds in the Schubert calculus of flags?
  7. Acknowledgements
  8. Bibliography