# 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
- Introduction
- Polynomial Systems
- Sparse Polynomial Systems
- The Polyhedral Homotopy Algorithm
- Real Solutions to Sparse Polynomial Systems

- Enumerative Real Algebraic Geometry
- Enumerative Real Algebraic Geometry
- Problems not involving general conditions
- The Stewart-Gough Platform
- Real rational cubics through 8 points in
**R**^{2}
- Common tangent lines to Spheres in
**R**^{n}

- Schubert Calculus
- The Schubert Calculus of Lines
- The Special Schubert Calculus
- Further Extensions of the Schubert Calculus

- The Conjecture of Shapiro and Shapiro
- The Conjecture of Shapiro and Shapiro for Grassmannians
- Rational functions with real critical points
- Extensions of the Conjecture of Shapiro

- Lower Bounds in the Schubert calculus
- Real Degree of Grassmann varieties
- Lower bounds in the Schubert calculus of flags?

- Acknowledgements
- Bibliography