**University Lecture Series**

Volume: 57;
2011;
200 pp;
Softcover

MSC: Primary 14;
Secondary 12

Print ISBN: 978-0-8218-5331-3

Product Code: ULECT/57

List Price: $50.00

AMS Member Price: $40.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-1652-2
Product Code: ULECT/57.E**

List Price: $50.00

AMS Member Price: $40.00

MAA Member Price: $45.00

#### Supplemental Materials

# Real Solutions to Equations from Geometry

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*Frank Sottile*

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.

#### Readership

Graduate students and research mathematicians interested in real algebraic geometry.

#### Reviews & Endorsements

... 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 computer-generated figures. His book offers a fresh, visual and colorful approach to real algebraic geometry.

-- Mathematical Reviews

...a very well-written 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

#### Table of Contents

# Table of Contents

## Real Solutions to Equations from Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Overview 112 free
- Real solutions of univariate polynomials 1324
- Sparse polynomial systems 2536
- Toric degenerations and Kushnirenko’s theorem 3748
- Fewnomial upper bounds 4960
- Fewnomial upper bounds from Gale dual polynomial systems 6172
- Lower bounds for sparse polynomial systems 7788
- Some lower bounds for systems of polynomials 91102
- Enumerative real algebraic geometry 105116
- The Shapiro Conjecture for Grassmannians 121132
- The Shapiro Conjecture for rational functions 133144
- Proof of the Shapiro Conjecture for Grassmannians 147158
- Beyond the Shapiro Conjecture for the Grassmannian 161172
- The Shapiro Conjecture beyond the Grassmannian 173184
- Bibliography 189200
- Index of notation 195206 free
- Index 197208 free
- Back Cover Back Cover1214