A classic problem in mathematics is solving systems of polynomial equations in
several unknowns. Today, polynomial models are ubiquitous and widely used
across the sciences. They arise in robotics, coding theory, optimization,
mathematical biology, computer vision, game theory, statistics, and numerous
other areas. This book furnishes a bridge across mathematical disciplines and
exposes many facets of systems of polynomial equations. It covers a wide
spectrum of mathematical techniques and algorithms, both symbolic and
numerical.

The set of solutions to a system of polynomial equations is an algebraic
variety—the basic object of algebraic geometry. The algorithmic study of
algebraic varieties is the central theme of computational algebraic geometry.
Exciting recent developments in computer software for geometric calculations
have revolutionized the field. Formerly inaccessible problems are now
tractable, providing fertile ground for experimentation and conjecture.

The first half of the book gives a snapshot of the state of the art of
computational algebraic geometry, i.e., of the algorithmic study of algebraic
varieties. Familiar themes covered in the first five chapters include
polynomials in one variable, Gröbner bases of zero-dimensional ideals,
Newton polytopes and Bernstein's Theorem, multidimensional resultants, and
primary decomposition.

The second half of the book explores polynomial equations from a variety of
novel and unexpected angles. It introduces interdisciplinary connections,
discusses highlights of current research, and outlines possible future
algorithms. Topics include computation of Nash equilibria in game theory,
semidefinite programming and the real Nullstellensatz, the algebraic geometry
of statistical models, the piecewise-linear geometry of valuations and amoebas,
and the Ehrenpreis-Palamodov theorem on linear partial differential equations
with constant coefficients.

Throughout the text, there are many hands-on examples and exercises,
including short but complete sessions in Maple®, MATLAB®, Macaulay
2, Singular, PHCpack, SOSTools, and CoCoA. These examples will
be particularly useful for readers with no background in algebraic geometry or
commutative algebra. Within minutes, readers can learn how to type in
polynomial equations and actually see some meaningful results on their computer
screens.

Readership

Graduate students and research mathematicians interested in
computational algebra and its applications.