On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED
Bachir El Khadir
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引用次数: 2

Abstract

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.
凸形式的平方和表示与广义Cauchy-Schwarz不等式
一个大于1度的凸形式总是非负的,因为它在原点与它的梯度一起消失。2007年,Parrilo提出了凸形式是否总是平方和的问题。几年后,Blekherman以否定的方式回答了这个问题,他通过体积论证表明,对于足够多的变量,一定存在低至4的非平方和的凸形式。值得注意的是,迄今为止还没有已知的例子。本文证明了所有4变量的4次凸形式都是平方和。我们还证明,如果与所谓的Cayley-Bacharach关系有关的Blekherman的一个猜想是正确的,那么同样的命题也适用于3变量和6次的凸形式。这是两种最小的情况,在这种情况下,人们可能有希望看到不是平方和的凸形式(由于已知的障碍物)。证明的一个主要组成部分是推导某些“广义Cauchy-Schwarz不等式”,这可能是独立的兴趣。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
0.00%
发文量
19
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