Preface
Chapter 1．Overview
1．1．Introduction
1．2．Polyhedral bounds
1．3．Upper bounds
1．4．The Wronski map and the Shapiro Conjecture
1．5．Lower bounds

Chapter 2．Real Solutions to Univariate Polynomials
2．1．Descartes's rule of signs
2．2．Sturm's Theorem
2．3．A topological proof of Sturm's Theorem

Chapter 3．Sparse Polynomial Systems
3．1．Polyhedral bounds
3．2．Geometric interpretation of sparse polynomial systems
3．3．Proof of Kushnirenko's Theorem
3．4．Facial systems and degeneracies

Chapter 4．Torie Degenerations and Kushnirenko's Theorem
4．1．Kushnirenko's Theorem for a simplex
4．2．Regular subdivisions and toric degenerations
4．3．Kushnirenko's Theorem via toric degenerations
4．4．Polynomial systems with only real solutions

Chapter 5．Fewnomial Upper Bounds
5．1．Khovanskii's fewnomial bound
5．2．Kushnirenko's Conjecture
5．3．Systems supported on a circuit

Chapter 6．Fewnomial Upper Bounds from Gale Dual Polynomial Systems
6．1．Gale duality for polynomial systems
6．2．New fewnomial bounds
6．3．Dense fewnomials

Chapter 7．Lower Bounds for Sparse Polynomial Systems
7．1．Polynomial systems as fibers of maps
7．2．Orientability of real toric varieties
7．3．Degree from foldable triangulations
7．4．Open problems

Chapter 8．Some Lower Bounds for Systems of Polynomials
8．1．Polynomial systems from posets
8．2．Sagbi degenerations
8．3．Incomparable chains， factoring polynomials， and gaps

Chapter 9．Enumerative Real Algebraic Geometry
9．1．3264 real conics
9．2．Some geometric problems
9．3．Schubert Calculus

Chapter 10．The Shapiro Conjecture for Grassmannians
10．1．The Wronski map and Schubert Calculus
10．2．Asymptotic form of the Shapiro Conjecture
10．3．Grassmann duality

Chapter 11．The Shapiro Conjecture for Rational Functions
11．1．Nets of rational functions
11．2．Schubert induction for rational functions and nets
11．3．Rational functions with prescribed coincidences

Chapter 12．Proof of the Shapiro Conjecture for Grassmannians
12．1．Spaces of polynomials with given Wronskian
12．2．The Gaudin model
12．3．The Bethe Ansatz for the Gaudin model
12．4．Shapovalov form and the proof of the Shapiro Conjecture

Chapter 13．Beyond the Shapiro Conjecture for the Grassmannian
13．1．Transversality and the Discriminant Conjecture
13．2．Maximally inflected curves
13．3．Degree of Wronski maps and beyond
13．4．The Secant Conjecture

Chapter 14．The Shapiro Conjecture Beyond the Grassmannian
14．1．The Shapiro Conjecture for the orthogonal Grassmannian
14．2．The Shapiro Conjecture for the Lagrangian Grassmannian
14．3．The Shapiro Conjecture for flag manifolds
14．4．The Monotone Conjecture
14．5．The Monotone Secant Conjecture
Bibliography
Index of Notation
Index