Relevant results on convex geometry are collected together in the appendix. We show by an example that his outline has gaps regarding isolated, singular points of the curve. Kippenhahn discovered this important statement of matrix theory and outlined its proof. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. The numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Hence this book might serve as an accessible introduction to current algebraic geometry. branch of mathematics like in the example 'geometric analysis' or analytic geometry. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Thomas Unger is Associate Professor at University College Dublin. His primary research interests are real algebraic geometry and convex algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Real Algebra is a graduate introduction to algebra and geometry over ordered fields, covering sums of squares (Hilberts 17th problem), the real spectrum. Toric varieties are here treated as complex analytic spaces.
It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985.
This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces.
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