Applications of Polynomial Systems

by

David A. Cox
Amherst College

The Book

Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert.

Examples in the book include oil wells, HIV infection, phylogenetic models, four-bar mechanisms, border rank, font design, Stewart-Gough platforms, rigidity of edge graphs, Gaussian graphical models, geometric constraint systems, and enzymatic cascades. The reader will encounter geometric objects such as Bézier patches, Cayley-Menger varieties, and toric varieties; and algebraic objects such as resultants, Rees algebras, approximation complexes, matroids, and toric ideals. Two important subthemes that appear in multiple chapters are toric varieties and algebraic statistics. The book also discusses the history of elimination theory, including its near elimination in the middle of the twentieth century.

The main goal is to inspire the reader to learn about the topics covered in the book. With this in mind, the book has an extensive bibliography containing over 350 books and papers.

Contents of the Book

The book consists of five chapters:

• Chapter 1: Elimination Theory
• Chapter 2: Numerical Algebraic Geometry
• Chapter 3: Geometric Modeling
• Chapter 4: Rigidity Theory
• Chapter 5: Chemical Reaction Networks

The Book has been Published

The book has been published by the American Mathematical Society as Number 134 of their CBMS Regional Conference Series in Mathematics. Click here for the AMS page for the book.

Typographical Errors

Here are the typographical errors known as of May 21, 2020:

• Page 8, line -2: "1892" should be "1907".
• Page 187, fourth display: "S+I → I → R" should be "S+I → 2I, I → R" (with α over the first arrow and β over the second).
• Page 211, line -15: "intepreted" should be "interpreted".
• Page 227, entry [45]: The entry should end with a period, not a comma.
• Page 237, entry [281]: "1909" should be "1907".

Contacting the Author

You can contact the author at the following email address: