革兰氏谱面的尺寸
Dimensions of faces of Gram spectrahedra
摘要：
设$f\inΣ_{n,2d}$为平方和。$f$的格拉姆谱面体是一个紧凑的凸集，它将$f$的所有平方和表示参数化。设$F\subseteq\mathrm{Gram}(f)$为其格拉姆谱面体的一个面。我们对$F$维数的上界感兴趣。我们表明，这个上界可以确定组合。事实证明，如果次数足够大，则实现该界限的面是革兰氏谱面体的面，使得形式$f$是奇异的。因此，我们也有兴趣找到更好的界限时，形式$f$是光滑的。
英文摘要：
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.
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