对方程组的实数解的理解、求解甚至仅仅确定解的存在性都是一个非常困难的问题，并且在数学以外的领域有着诸多应用。尽管总体上我们不抱太大的希望，但令人惊喜的是，我们发现相当一部分拥有额外结构的方程组常常与几何相关。
弗兰克·索蒂尔著的《方程组实数解的几何方法（英文版）（精）》重点讨论基于环簇和Grassmann流形构建的方程组。这是由于不仅这些理论为人们所熟知，而且所涉及的方程组在应用中常见。全书共分三个主题：实数解个数的上界、实数解个数的下界、所有解均为实数的方程组的几何问题。本书首先给出一个概述，包括单变量多项式方程组的实数解以及稀疏多项式方程组的几何结构的背景知识；前半部分讲述稀疏多项式方程组的“少”项式（fewnomial）上界及下界；后半部分先选取了一些所有解均为实数的方程组的几何问题，然后在最后五章介绍Shapiro猜想，其中相关的多项式方程组只有实数解。
本书适合于对实代数几何感兴趣的研究生和专业研究人员阅读。

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