viii CONTENTS

Chapter 8. Some Lower Bounds for Systems of Polynomials 91

8.1. Polynomial systems from posets 91

8.2. Sagbi degenerations 96

8.3. Incomparable chains, factoring polynomials, and gaps 100

Chapter 9. Enumerative Real Algebraic Geometry 105

9.1. 3264 real conics 105

9.2. Some geometric problems 109

9.3. Schubert Calculus 116

Chapter 10. The Shapiro Conjecture for Grassmannians 121

10.1. The Wronski map and Schubert Calculus 122

10.2. Asymptotic form of the Shapiro Conjecture 124

10.3. Grassmann duality 130

Chapter 11. The Shapiro Conjecture for Rational Functions 133

11.1. Nets of rational functions 133

11.2. Schubert induction for rational functions and nets 137

11.3. Rational functions with prescribed coincidences 141

Chapter 12. Proof of the Shapiro Conjecture for Grassmannians 147

12.1. Spaces of polynomials with given Wronskian 148

12.2. The Gaudin model 152

12.3. The Bethe Ansatz for the Gaudin model 154

12.4. Shapovalov form and the proof of the Shapiro Conjecture 157

Chapter 13. Beyond the Shapiro Conjecture for the Grassmannian 161

13.1. Transversality and the Discriminant Conjecture 161

13.2. Maximally inflected curves 164

13.3. Degree of Wronski maps and beyond 167

13.4. The Secant Conjecture 170

Chapter 14. The Shapiro Conjecture Beyond the Grassmannian 173

14.1. The Shapiro Conjecture for the orthogonal Grassmannian 173

14.2. The Shapiro Conjecture for the Lagrangian Grassmannian 175

14.3. The Shapiro Conjecture for flag manifolds 179

14.4. The Monotone Conjecture 180

14.5. The Monotone Secant Conjecture 186

Bibliography 189

Index of Notation 195

Index 197