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
Previous Page Next Page