Softcover ISBN:  9780821853313 
Product Code:  ULECT/57 
List Price:  $53.00 
MAA Member Price:  $47.70 
AMS Member Price:  $42.40 
Electronic ISBN:  9781470416522 
Product Code:  ULECT/57.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $40.00 

Book DetailsUniversity Lecture SeriesVolume: 57; 2011; 200 ppMSC: Primary 14; Secondary 12;
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a difficult problem with many applications outside of mathematics. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure often coming from geometry.
This book focuses on equations from toric varieties and Grassmannians. Not only is much known about these, but such equations are common in applications. There are three main themes: upper bounds on the number of real solutions, lower bounds on the number of real solutions, and geometric problems that can have all solutions be real. The book begins with an overview, giving background on real solutions to univariate polynomials and the geometry of sparse polynomial systems. The first half of the book concludes with fewnomial upper bounds and with lower bounds to sparse polynomial systems. The second half of the book begins by sampling some geometric problems for which all solutions can be real, before devoting the last five chapters to the Shapiro Conjecture, in which the relevant polynomial systems have only real solutions.ReadershipGraduate students and research mathematicians interested in real algebraic geometry.

Table of Contents

Chapters

Chapter 1. Overview

Chapter 2. Real solutions of univariate polynomials

Chapter 3. Sparse polynomial systems

Chapter 4. Toric degenerations and Kushnirenko’s theorem

Chapter 5. Fewnomial upper bounds

Chapter 6. Fewnomial upper bounds from Gale dual polynomial systems

Chapter 7. Lower bounds for sparse polynomial systems

Chapter 8. Some lower bounds for systems of polynomials

Chapter 9. Enumerative real algebraic geometry

Chapter 10. The Shapiro Conjecture for Grassmannians

Chapter 11. The Shapiro Conjecture for rational functions

Chapter 12. Proof of the Shapiro Conjecture for Grassmannians

Chapter 13. Beyond the Shapiro Conjecture for the Grassmannian

Chapter 14. The Shapiro Conjecture beyond the Grassmannian


Additional Material

Reviews

... I am convinced that this book can be a source of inspiration for newcomers to real algebraic geometry, as well as a timely update of the latest results for more experienced scholars. I am particularly impressed by the author's ability to convey visually some technical ideas with the help of splendid computergenerated figures. His book offers a fresh, visual and colorful approach to real algebraic geometry.
Mathematical Reviews 
...a very wellwritten book that discusses some very exciting and modern algebraic geometry that has roots in questions that can be easily formulated even at the level of high school students. While the book gets quite technical at times, Sottile manages to include many examples and pictures to keep the exposition clear and light. ... I learned quite a bit from the book and I would recommend it to those looking to learn more about the subject.
MAA Reviews


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 Get Permissions
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a difficult problem with many applications outside of mathematics. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure often coming from geometry.
This book focuses on equations from toric varieties and Grassmannians. Not only is much known about these, but such equations are common in applications. There are three main themes: upper bounds on the number of real solutions, lower bounds on the number of real solutions, and geometric problems that can have all solutions be real. The book begins with an overview, giving background on real solutions to univariate polynomials and the geometry of sparse polynomial systems. The first half of the book concludes with fewnomial upper bounds and with lower bounds to sparse polynomial systems. The second half of the book begins by sampling some geometric problems for which all solutions can be real, before devoting the last five chapters to the Shapiro Conjecture, in which the relevant polynomial systems have only real solutions.
Graduate students and research mathematicians interested in real algebraic geometry.

Chapters

Chapter 1. Overview

Chapter 2. Real solutions of univariate polynomials

Chapter 3. Sparse polynomial systems

Chapter 4. Toric degenerations and Kushnirenko’s theorem

Chapter 5. Fewnomial upper bounds

Chapter 6. Fewnomial upper bounds from Gale dual polynomial systems

Chapter 7. Lower bounds for sparse polynomial systems

Chapter 8. Some lower bounds for systems of polynomials

Chapter 9. Enumerative real algebraic geometry

Chapter 10. The Shapiro Conjecture for Grassmannians

Chapter 11. The Shapiro Conjecture for rational functions

Chapter 12. Proof of the Shapiro Conjecture for Grassmannians

Chapter 13. Beyond the Shapiro Conjecture for the Grassmannian

Chapter 14. The Shapiro Conjecture beyond the Grassmannian

... I am convinced that this book can be a source of inspiration for newcomers to real algebraic geometry, as well as a timely update of the latest results for more experienced scholars. I am particularly impressed by the author's ability to convey visually some technical ideas with the help of splendid computergenerated figures. His book offers a fresh, visual and colorful approach to real algebraic geometry.
Mathematical Reviews 
...a very wellwritten book that discusses some very exciting and modern algebraic geometry that has roots in questions that can be easily formulated even at the level of high school students. While the book gets quite technical at times, Sottile manages to include many examples and pictures to keep the exposition clear and light. ... I learned quite a bit from the book and I would recommend it to those looking to learn more about the subject.
MAA Reviews