Geometry of hyperbolic polynomials

双曲多项式的几何

• 批准号: 426054364
• 负责人: Professor Dr. Daniel Plaumann
• 金额: --
• 依托单位: Abteilung Algebra
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/426054364?language=en
• 关键词: Geometry; hyperbolic; polynomials

At the heart of this project are hyperbolic polynomials and their hyperbolicity cones. These are real polynomials in several variables bounding a convex cone, with a particular reality condition on the roots. They originate in complex analysis and the theory of partial differential equations, but have more recently arisen in optimization, combinatorics, and probability theory. Hyperbolic polynomials can be seen as generalizations of characteristic polynomials of real symmetric matrix pencils. On the one hand, one can try to express them as such (determinantal representations) and compare hyperbolicity cones to the resulting cones of matrices. It is an open question whether this is always possible in a certain sense. On the other hand, one can also imitate matrix theory within the framework of hyperbolicity. This project will focus on the latter and study hyperbolicity cones from the point of view of convex algebraic geometry, with a particular emphasis on the interplay of duality theories, namely (1) duality of algebraic varieties in projective space, (2) duality of convex cones, and (3) convex programming duality. The last is well understood and heavily exploited in semidefinite programming, but remains largely incomplete for hyperbolic programming. Along with these fundamental questions, we will also apply such techniques to concrete problems supplied by the above connections to other fields.

这个项目的核心是双曲多项式和它们的双曲锥。这些都是真实的多项式在几个变量的边界凸锥，与一个特定的现实条件的根源。它们起源于复分析和偏微分方程理论，但最近出现在优化、组合学和概率论中。双曲多项式可以看作是真实的对称矩阵束的特征多项式的推广。一方面，我们可以试着将它们表达为（行列式表示），并将双曲锥与矩阵的结果锥进行比较。在某种意义上，这是否总是可能的，这是一个悬而未决的问题。另一方面，人们也可以在双曲性的框架内模仿矩阵理论。本项目将侧重于后者，并从凸代数几何的角度研究双曲锥，特别强调对偶理论的相互作用，即（1）射影空间中代数簇的对偶，（2）凸锥的对偶，以及（3）凸规划对偶。最后一个在半定规划中得到了很好的理解和大量的利用，但在双曲规划中仍然很不完整。沿着这些基本问题，我们也将把这些技术应用于上述与其他领域的联系所提供的具体问题。