Geometry of hyperbolic polynomials

这个项目的核心是双曲多项式及其双曲性圆锥。它们是多个变量的实多项式，其根部具有特定的实值条件，并以凸锥为边界。它们起源于复杂分析和偏微分方程理论，但最近出现在最优化、组合学和概率论中。双曲多项式可以看作是实对称矩阵铅笔特征多项式的推广。一方面，人们可以尝试将它们表示为行列式表示，并将双曲性圆锥与所得到的矩阵圆锥进行比较。这在某种意义上是否总是可能的，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。本课题将集中于后者，从凸代数几何的角度研究双曲性锥，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性，(2)凸锥的对偶，和(3)凸规划对偶。后者在半定规划中得到了很好的理解和充分利用，但在双曲规划中仍有很大程度的不完善。除了这些基本问题外，我们还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。

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.