1/17/2024 0 Comments Convex algebraic geometry![]() ![]() This part of the book can thus serve for a one-semester introduction to algebraic geometry, with the first part serving as a reference for combinatorial geometry. Algorithmic and quantitative real algebraic geometry, DIMACS Series in. 705: 2011: Introducing SOSTOOLS: A general purpose sum of squares programming solver. Many of the general concepts of algebraic geometry arise in this treatment and can be dealt with concretely. Semidefinite Optimization and Convex Algebraic Geometry. The second part introduces toric varieties in an elementary way, building on the concepts of combinatorial geometry introduced in the first part. The emerging eld of Convex Algebraic Geometry originates from a natural coalescence of ideas in Convex Geometry, Optimization, and Algebraic Geometry, and has witnessed great progress over the last few years, see BPT13 for surveys. ![]() The central objects of study in this rapidly developing field are convex sets with algebraic structure. This part also provides large parts of the mathematical background of linear optimization and of the geometrical aspects in Computer Science. Convex algebraic geometry is an evolving subject area arising from a synthesis of ideas and techniques from optimization, convex geometry, and algebraic geometry. Since the discussion here is independent of any applications to algebraic geometry, it would also be suitable for a course in geometry. The fist part of the book contains an introduction to the theory of polytopes - one of the most important parts of classical geometry in n-dimensional Euclidean space. This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (or torus embeddings). The numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Contents: Convex Bodies.- Combinatorial Theory of Polytopes and polyhedral sets.- Polyhedral spheres.- Minkowski sum and mixed volume.- Lattice Polytopes and fans.- Toric Varieties.- Sheaves and projective toric varieties.- Cohomology of toric varieties.Bibliography Includes bibliographical references (p. ![]()
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