论文标题
SOS锥边界中的严格阳性多项式
Strictly positive polynomials in the boundary of the SOS cone
论文作者
论文摘要
我们研究可分解为真实多项式总和的真实多项式锥的边界。该锥体包含在非负多项式的锥中，并且两个锥体共享其边界的一部分，这对应于至少一个点消失的多项式。我们专注于未共享边界的一部分，对应于严格的正多项式。 对于3个变量中6度的多项式和4个变量中的4度，该边界的特征是G. blekherman的特征。对于更多变量或更高程度的情况，G。Blekherman，R。Sinn和M. Velasco的结果以及其他基于一般猜想的作者的结果为最大多项式数量的界限提供了界限，这些多项式可以在SOS分解中出现，并且在Gram Spectrahedron中的矩阵最大等级。结合了理论结果和计算技术，我们计算了示例，这些示例使我们能够证明所有程度和变量数量的边界的最佳性。此外，我们给出了以下问题的示例：锥体边界中的示例，其总和小于$ n $正方形，并且具有共同的复杂根，以及边界中多项式的示例，长度大于尺寸的预期。
We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of more variables or higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on general conjectures give bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than $n$ squares and have common complex roots, and examples of polynomials in the boundary with length larger than the expected from the dimension.
