凸形的平方和表示与广义柯西-施瓦茨不等式
On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities
摘要：
度大于1的凸形式总是非负的，因为它在原点与梯度一起消失。2007年，Parrilo提出了凸形式是否总是平方和的问题。几年后，Blekherman回答了这个问题的负面显示，通过大量的参数，为足够高的变量，必须有凸形式的程度低至4，而不是平方和。值得注意的是，迄今为止还没有已知的例子。本文证明了4元4次凸形式都是平方和。我们还表明，如果一个猜想的Blekherman有关所谓的凯莱-Bacharach关系是真的，那么同样的陈述成立凸形式的3个变量和度6。这是两个最小的情况下，人们将有任何希望看到凸形式，而不是平方和（由于已知的障碍）。证明的一个主要成分是某些“广义柯西-施瓦茨不等式“的推导，这可能是独立的利益。
英文摘要：
A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized Cauchy-Schwarz inequalities" which could be of independent interest.
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