Sums of squares in convex algebraic geometry

借助于线性规划，可以有效地优化多面体上的线性多项式。自20世纪40年代以来，这项技术基本上已经广为人知，并有无数的应用。自20世纪80年代以来，人们知道半定规划也可以用一种有效的方法来求解。它们是线性规划的一个深远的推广，并有许多重要的应用，例如在电气工程中。半定程序在对称矩阵的线性束中的半正定矩阵上优化线性目标函数。哪些问题可以被建模为半定规划的问题是至关重要的。这就引出了刻画半定可表示集的问题。人们可以很容易地看到，任何这样的集合都是凸的和半代数的。相反，许多重要的凸半代数集都有半定表示。多年来，人们一直猜测每个凸半代数集都有一个半定表示(即所谓的Helton-Nie猜想)。这个猜想最近被项目负责人证明是错误的。借助于一个新的刻画，构造了一类显式反例。这种刻画是基于用平方和表示正多项式的。然而，在具体情况下，要确定这一条件是否成立往往很麻烦。这就是为什么需要能够以更灵活的方式处理的替代标准。此外，通过对Helton-Nie猜想的反证，出现了各种新的未决问题。这个项目的主要目标之一是分离出这样灵活的半定表示标准，并至少回答一些开放的问题。另一个具有理论和实践意义的问题是给定集合的半定表示的复杂性。总体而言，人们对此知之甚少。然而，对于曲线的凸壳，具体的结果似乎是可能的，这是该项目的另一个主要目标。与第一个目标一样，平方和表示法也是这个问题的中心，问题是给出求和次数的上界。第三个子项目研究给定多项式的所有平方和表示的族。它们形成了一个紧凑的凸体，其结构到目前为止还知之甚少。新技术的使用有望在这方面带来重大进步。

With the help of linear programming it is possible to optimize linear polynomials over polyhedra in an effective way. This technique is basically well-known since the 1940s and has countless applications. Since the 1980s it is known that also semidefinite programs can be solved in an effective way. They are a far reaching generalization of linear programs and have many important applications, e.g. in electrical engineering. A semidefinite program optimizes a linear objective function over the positive semidefinite matrices in a linear pencil of symmetric matrices. The question which problems can be modelled as semidefinite programs is of basic importance. It leads to the problem of characterizing the semidefinitely representable sets. One can easily see that any such set is convex and semi-algebraic. Conversely, plenty of important classes of convex semi-algebraic sets are known to have semidefinite representations. For many years it was therefore conjecture that every convex semi-algebraic set should have a semidefinite representation (so-called Helton-Nie conjecture). This conjecture was recently shown to be false by the project leader. With the help of a new characterization, classes of explicit counter-examples were constructed. This characterization is based on representations of positive polynomials by sums of squares. However, in concrete cases it is often cumbersome to decide whether this condition holds. This is why alternative criteria are needed that can be handled in a more flexible way. Besides, a variety of new open question has emerged through the disproof of the Helton-Nie conjecture. One of the main goals of this project is to isolate such flexible criteria for semidefinite representability, and to answer at least some of the open questions. Another question of theoretical and practical importance asks for the complexity of semidefinite representations of given sets. Very little is known here in general. For convex hulls of curves however, concrete results seem possible, which is another major goal of the project. As with the first goal, sums of squares representations are at the center of this question as well, and the problem is to give upper bounds for the degrees of the summands. A third subproject studies the family of all sums of squares representations of a given polynomial. They form a compact convex body about whose structure only little is known so far. The use of new techniques is expected to bring a significant advance here.