Do Sums of Squares Dream of Free Resolutions?
Do Sums of Squares Dream of Free Resolutions?
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平方和梦想自由解决方案吗？
DOI：
10.1137/16m1084560
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Mauricio Velasco
中科院分区：
文献类型：
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作者：
Grigoriy Blekherman;Rainer Sinn;Mauricio Velasco
我们将实射影变量$X$联想到实代数几何中的两个基本凸锥：二次型的锥$P_X$在$X$上是非负的，以及线性形式的平方和的锥$\Sigma_X$。偶锥$\Sigma_X^\ast$是一个谱面体，我们证明了它的凹凸性与$X$的同调性密切相关。例如，当且仅当X具有Castelnuovo-Mumford正则性2时，我们证明了$\Sigma_X^\ast$的所有极值射线的秩为1。更一般地说，如果$\Sigma_X^\ast$有秩为$p > 1$的极值射线，则$X$不满足属性$N_{2,p}$。我们证明了在许多情况下逆命题也成立：对于性质$N_{2,p}$不成立的最小$p$等于大于1的$\Sigma_X^\ast$的极值射线的最小秩。这些结果使我们可以将Blekherman-Smith-Velasco关于非负多项式和平方和的等式的工作从不可约变型推广到约化格式，并对所有只有一级极值射线的光谱锥进行分类。我们的结果也适用于正半定矩阵补全问题和投影变换上的截断矩问题。
We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $\Sigma_X$ of sums of squares of linear forms. The dual cone $\Sigma_X^\ast$ is a spectrahedron and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $\Sigma_X^\ast$ have rank one if and only if X has Castelnuovo-Mumford regularity two. More generally, if $\Sigma_X^\ast$ has an extreme ray of rank $p > 1$, then $X$ does not satisfy the property $N_{2,p}$. We show that the converse also holds in a wide variety of situations: the smallest $p$ for which property $N_{2,p}$ does not hold is equal to the smallest rank of an extreme ray of $\Sigma_X^\ast$ greater than one. These results allow us to generalize the work of Blekherman-Smith-Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and to classify all spectrahedral cones with only rank one extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.