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。这种技术自20世纪40年代以来基本上是众所周知的，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求解。它们是线性规划的广泛推广，并有许多重要的应用，例如在电气工程中。半定规划优化线性目标函数在对称矩阵线性束中的半正定矩阵上。哪些问题可以建模为半定规划是一个基本的重要问题。这就引出了半定可表示集的刻画问题。我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，已知许多重要的凸半代数集类具有半定表示。多年来，因此猜想，每一个凸半代数集应该有一个半定表示（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。借助一个新的刻画，构造了一类显式反例。这种表征是基于正多项式的平方和表示。然而，在具体情况下，决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，还出现了许多新的问题。这个项目的主要目标之一是隔离半定可表示性的灵活标准，并回答至少一些开放的问题。另一个问题的理论和实践的重要性要求的复杂性半定表示的给定集。一般来说，这里知之甚少。然而，对于曲线的凸包，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，平方和表示也是这个问题的中心，问题是给出被加数的次数的上界。第三个子项目研究一个给定多项式的所有平方和表示的家庭。它们形成一个紧凑的凸体，其结构至今知之甚少。新技术的使用有望给这里带来重大进步。

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.