Cones between the cones of positive semidefinite forms and sums of squares

IF 0.5 4区 数学 Q3 MATHEMATICS
Charu Goel, Sarah Hess, Salma Kuhlmann
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引用次数: 0

Abstract

For n, d ∈ ℕ, the cone 𝓟 n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σ n+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n+1,2d and 𝓟 n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V kn ⊊ … ⊊ Vn ⊊ … ⊊ V 0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V 0, …, V n (and V n+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n+1,2d . We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.
介于正半定式和平方和之间的锥体
对于 n, d∈ ℕ,度数为 2d 的 n + 1 变数中正半定实数形式的锥𝓟 n+1,2d 包含可表示为实数形式有限平方和的子锥Σ n+1,2d 。希尔伯特[11] 证明了这些锥体在希尔伯特情形 (n + 1, 2d) 中完全重合,即 n + 1 = 2 或 2d = 2 或 (n + 1, 2d) = (3, 4)。在本文中,我们通过[4]中的格拉姆矩阵方法,在不可还原的投影变种 V k-n ⊊ ... ⊊ Vn ⊊ ... ⊊ V 0 的滤波上,诱导出介于 Σ n+1,2d 和 𝓟 n+1,2d 之间的中间锥的滤波,其中包含维罗纳变种。通过证明 V 0,...,V n(以及当 n = 2 时的 V n+1)是最小度的变项,我们证明了相应的中间锥与Σ n+1,2d 重合。此外,我们还证明,在 n ≥ 3 的 (n + 1)-ary 四元数和 n ≥ 2 的 (n + 1)-ary 六元数的非希尔伯特情况下,所有剩余的圆锥内含都是严格的。
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
Advances in Geometry
Advances in Geometry 数学-数学
CiteScore
1.00
自引率
0.00%
发文量
31
审稿时长
>12 weeks
期刊介绍: Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.
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