实代数几何教程：正性与平方和
This textbook is designed for a one-year graduate course in real algebraic geometry, with a particular focus on positivity and sums of squares of polynomials. The first half of the book features a thorough introduction to ordered fields and real closed fields, including the Tarski–Seidenberg projection theorem and transfer principle. Classical results such as Artin's solution to Hilbert's 17th problem and Hilbert's theorems on sums of squares of polynomials are presented in detail. Other features include careful introductions to the real spectrum and to the geometry of semialgebraic sets. The second part studies Archimedean positivstellensätze in great detail and in various settings, together with important applications. The techniques and results presented here are fundamental to contemporary approaches to polynomial optimization. Important results on sums of squares on projective varieties are covered as well. The last part highlights applications to semidefinite programming and polynomial optimization, including recent research on semidefinite representation of convex sets. Written by a leading expert and based on courses taught for several years, the book assumes familiarity with the basics of commutative algebra and algebraic varieties, as can be covered in a one-semester first course. Over 350 exercises, of all levels of difficulty, are included in the book.
English | PDF（True）| 2024 | 411页| ISBN:3031692128 | 6.5 MB这本教科书是为一年的实代数几何研究生课程设计的，特别关注多项式的正和平方和。本书的前半部分详细介绍了有序场和实闭场，包括塔尔斯基-塞登伯格投影定理和传递原理。详细介绍了经典结果，如Artin对Hilbert第17问题的解和Hilbert关于多项式平方和的定理。其他特征包括仔细介绍实谱和半代数集的几何。第二部分详细研究了阿基米德正态恒星，并在各种环境下进行了研究，以及重要的应用。这里介绍的技术和结果是当代多项式优化方法的基础。文中还讨论了射影变量平方和的重要结果。最后一部分重点介绍了半定规划和多项式优化的应用，包括凸集半定表示的最新研究。本书由一位权威专家撰写，基于多年教授的课程，假设读者熟悉交换代数和代数变体的基础知识，这可以在一学期的第一门课程中涵盖。这本书收录了350多个不同难度的练习。本站不对文件进行储存，仅提供文件链接，请自行下载，本站不对文件内容负责，请自行判断文件是否安全，如发现文件有侵权行为，请联系管理员删除。
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