Ebook: Applications of polynomial systems

Polynomials have been around for a long time. They play a prominent role
in many applications and are basic objects in commutative algebra and algebraic
geometry. In the 20th century, the foundations of algebraic geometry required a
level of abstraction that made these tools hard to use for many people interested in
applications. But with the advent of powerful algorithms, personal computers, and
the theory of sparse polynomials, we now have computational commutative algebra
and computational algebraic geometry. Coupled with accessible introductory texts,
recent years have witnessed a remarkable range of applications.
The goal of this volume is to introduce some of these applications. The five
chapters are based on the NSF-CBMS Regional Research Conference Applications
of Polynomial Systems held at Texas Christian University (TCU), June 4–8, 2018.
The format of the conference, replicated in this book, is that each day David Cox
gave two lectures on a topic, followed by a third lecture by an expert on the topic.
The book is not a complete introduction to the topics covered. Many proofs
are sketched or omitted, and the emphasis is on the examples. The hope is that
the brief presentation provided here inspires you to learn more.
The first two chapters set the background for the rest of the book. Chapter 1
focuses on the rich history of elimination theory from the work of Newton and
Bézout in the 17th and 18th centuries through the decline of elimination theory in
the middle of the 20th century, followed by the emergence of symbolic computation
in the latter part of the century. In the final section of the chapter, Carlos D’Andrea
brings elimination theory into the 21st century.
Chapter 2 begins with the tension between the perfect information needed
for symbolic computation and the messiness of the information coming from the
real world. The chapter explores linear algebra and homotopy continuation, two
commonly used methods for solving systems of polynomial equations numerically.
This is where we meet our first substantial applications. Jonathan Hauenstein
concludes the chapter with a discussion of sampling in numerical algebraic geometry.
The remaining chapters of the book focus on three substantial applications.
Chapter 3 explores the geometry and algebra of geometric modeling, with some
unexpected connections to toric varieties, algebraic statistics, and Rees algebras.
At the end of the chapter, Hal Schenck surveys a variety of algebraic tools which
are used in geometric modeling.
Chapter 4 is devoted to the geometry and combinatorics of rigidity theory.
For bar-and-joint frameworks, basic objects include graphs and rigidity matrices,
though polytopes and matroids also have an important role to play. A fun result
due to Gross and Sullivant uses the rigidity matroid to study maximum likelihood
estimates of Gaussian graphical models coming from planar graphs. Jessica Sidman
ends the chapter with a discussion of body-and-bar and body-and-cad frameworks
and the study of non-generic frameworks via the Grassmann-Cayley algebra.
Chapter 5 shifts the scene to chemical reaction networks, where the Law of
Mass Action leads to a lovely combination of graph theory and dynamical systems.
This framework includes examples from chemistry, biology, epidemiology, and pop-
ulation genetics. Algebraic geometry enters the picture because the steady states
are defined by a polynomial system. Toric varieties also have a role to play by the
pioneering work of Karin Gatermann. Alicia Dickenstein finishes the chapter with
an exposition of the interesting things that can happen in biochemical reaction
networks.
Each chapter starts at a relatively elementary level, with more advanced topics
introduced as needed. The algebraic geometry background can be found in the book
[104]. In addition to the main applications presented in Chapters 3, 4, and 5, the
twin themes of toric varieties and algebraic statistics play a prominent role in the
book. Readers unfamiliar with this material should consult the expository papers
[98] and [214].
We hope you enjoy the book! We had fun writing it.