论文标题
三元二次形式的权力边界等级
Border rank of powers of ternary quadratic forms
论文作者
论文摘要
我们在三个变量中确定任何二次形式的每个功率的边界等级。由于排名$ 1 $和排名$ 2 $二次形式的问题可以减少以确定二进制形式的权力等级，因此我们主要关注非分类二次形式。我们首先考虑二次表格$ q_ {n} = x_1^{2}+\ dots+x_n^{2} $在任意变量的$ n $中。我们确定了任何功率$ q_n^s $的可靠性理想，证明它与$ s+1 $的谐波多项式产生的同质理想相对应。使用此结果，我们选择了三个变量的二次形式的每种功率的Apolar理想中包含的特定理想，该变量在不失去一般性的情况下，我们假设是$ q_3 $的形式。验证某些属性后，我们利用边界极性的最新技术确定任何功率$ q_3^s $的边框等级等于其中间催化剂矩阵的等级，即$（s+1）（s+2）/2 $。
We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank $1$ and rank $2$ quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on non-degenerate quadratic forms. We begin by considering the quadratic form $q_{n}=x_1^{2}+\dots+x_n^{2}$ in an arbitrary number $n$ of variables. We determine the apolar ideal of any power $q_n^s$, proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree $s+1$. Using this result, we select a specific ideal contained in the apolar ideal for each power of a quadratic form in three variables, which, without loss of generality, we assume to be the form $q_3$. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power $q_3^s$ is equal to the rank of its middle catalecticant matrix, namely $(s+1)(s+2)/2$.
