论文标题
克谱的面孔的尺寸
Dimensions of faces of Gram spectrahedra
论文作者
论文摘要
令$ f \inς_{n，2d} $为正方形。 $ f $的革兰氏谱图是一个紧凑的凸集，该集合列出了$ f $的所有正方形表示总和。令$ f \ subseteq \ mathrm {gram}（f）$成为其革兰氏谱的面孔。我们对$ f $的尺寸感兴趣。我们表明，可以组合确定该上限。事实证明，如果学位足够大，那么意识到这一界限的面孔是克谱的面孔，使得$ f $的形式是单数的。因此，每当$ f $表单平滑时，我们也有兴趣找到更好的界限。
Let $f\inΣ_{n,2d}$ be a sum of squares. The Gram spectrahedron of $f$ is a compact, convex set that parametrizes all sum of squares representations of $f$. Let $F\subseteq\mathrm{Gram}(f)$ be a face of its Gram spectrahedron. We are interested in upper bounds for the dimension of $F$. We show that this upper bound can be determined combinatorially. As it turns out, if the degree is large enough, a face realizing this bound, is a face of a Gram spectrahedron such that the form $f$ is singular. Thus we are also interested in finding better bounds whenever the form $f$ is smooth.
