The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the well-known text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gr¨obner bases of zero-dimensional ideals, Newton polytopes and Bernstein’s Theorem, multidimensional resultants, and primary decomposition.

The second half of this book explores polynomial equations from a variety of novel and perhaps unexpected angles. Interdisciplinary connections are introduced, highlights of current research are discussed, and the author’s hopes for future algorithms are outlined. The topics in these chapters include computation of Nash equilibria in game theory, semidefinite programming and the real Nullstellensatz, the algebraic geometry of statistical models, the piecewiselinear geometry of valuations and amoebas, and the Ehrenpreis-Palamodov theorem on linear partial differential equations with constant coefficients.

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