Ideals, Varieties, and Algorithms

最新推荐文章于 2022-06-08 20:40:30 发布

mathsoperator

最新推荐文章于 2022-06-08 20:40:30 发布

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分类专栏：代数文章标签： variables object system methods

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本书深入浅出地介绍了代数几何的核心概念，包括希尔伯特基础定理、诺特尔定理、不变理论、射影几何和维数理论等。通过算法解答一元或多变量多项式方程组的问题，揭示了代数与几何之间的紧密联系。近四十年来，计算方法的复兴推动了理论进步和应用创新，例如在机器人学和几何定理证明中。

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Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?

The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

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