Contents Preface ix Chapter 1. Overview 1 1.1. Introduction 2 1.2. Polyhedral bounds 3 1.3. Upper bounds 4 1.4. The Wronski map and the Shapiro Conjecture 5 1.5. Lower bounds 8 Chapter 2. Real Solutions to Univariate Polynomials 13 2.1. Descartes’s rule of signs 13 2.2. Sturm’s Theorem 16 2.3. A topological proof of Sturm’s Theorem 19 Chapter 3. Sparse Polynomial Systems 25 3.1. Polyhedral bounds 26 3.2. Geometric interpretation of sparse polynomial systems 27 3.3. Proof of Kushnirenko’s Theorem 29 3.4. Facial systems and degeneracies 33 Chapter 4. Toric Degenerations and Kushnirenko’s Theorem 37 4.1. Kushnirenko’s Theorem for a simplex 37 4.2. Regular subdivisions and toric degenerations 39 4.3. Kushnirenko’s Theorem via toric degenerations 44 4.4. Polynomial systems with only real solutions 47 Chapter 5. Fewnomial Upper Bounds 49 5.1. Khovanskii’s fewnomial bound 49 5.2. Kushnirenko’s Conjecture 54 5.3. Systems supported on a circuit 56 Chapter 6. Fewnomial Upper Bounds from Gale Dual Polynomial Systems 61 6.1. Gale duality for polynomial systems 62 6.2. New fewnomial bounds 66 6.3. Dense fewnomials 74 Chapter 7. Lower Bounds for Sparse Polynomial Systems 77 7.1. Polynomial systems as fibers of maps 78 7.2. Orientability of real toric varieties 80 7.3. Degree from foldable triangulations 84 7.4. Open problems 89 vii
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