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

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