Dimensional Differences Between Faces of the Cones of Nonnegative Polynomials and Sums of Squares

被引:4
|
作者
Blekherman, Grigoriy [1 ]
Iliman, Sadik [2 ]
Kubitzke, Martina [2 ]
机构
[1] Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA
[2] Goethe Univ Frankfurt, FB Inst Math 12, D-60054 Frankfurt, Germany
基金
美国国家科学基金会;
关键词
D O I
10.1093/imrn/rnu202
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. The gaps occur generically, they are not the product of selecting special faces of the cones. For ternary forms and quaternary quartics, we characterize when these differences are observed. Moreover, we provide an explicit description for these differences in the two smallest cases, in which the cone of nonnegative polynomials and the cone of sums of squares are different. Our results follow from more general results concerning the relationship between the degree 2d component of the second symbolic power of the vanishing ideal of points in projective space and the square of the degree d component of the vanishing ideal.
引用
收藏
页码:8437 / 8470
页数:34
相关论文
共 50 条