AI 中文总结
解决欧几里得距离矩阵哈达玛幂秩恒等式的开放问题，通过展示核分解$D^{(n)} = \Phi_V\, M\, \Phi_V^T$，利用核的三项式展开表明$M$非奇异，从而完成秩恒等式。
AI 中文摘要
霍瓦特等人（《数学化学杂志》，2014年）表明，欧几里得距离矩阵的第n次哈达玛幂$D^{(n)}$的秩满足$\operatorname{rank}D^{(n)} \le R_d^n$，并证明当存在零化多项式时该不等式严格成立。而不存在零化多项式时$\operatorname{rank}D^{(n)} = R_d^n$这一逆命题留作开放问题。我们通过展示核分解$D^{(n)} = \Phi_V\, M\, \Phi_V^T$解决了该问题，其中$\Phi_V$是多项式空间$V$上的求值矩阵，$M$是与点配置无关的通用矩阵。核的三项式展开表明$M$具有块对角结构，其块是具有正系数的Gram矩阵之和，这导致$M$非奇异并完成了秩恒等式。
英文摘要
Horvat et al. (J. Math. Chem., 2014) showed that the rank of the $n$-th Hadamard power $D^{(n)}$ of a Euclidean distance matrix satisfies $\operatorname{rank}D^{(n)} \le R_d^n$, and proved that the inequality is strict whenever an annihilating polynomial exists. The converse - that the absence of annihilating polynomials forces $\operatorname{rank}D^{(n)} = R_d^n$ - was left as an open problem. We resolve it by exhibiting a kernel factorisation $D^{(n)} = Φ_V\, M\, Φ_V^T$, where $Φ_V$ is the evaluation matrix on the polynomial space $V$ and $M$ is a universal matrix independent of the point configuration. A trinomial expansion of the kernel reveals that $M$ has a block-diagonal structure whose blocks are sums of Gram matrices with positive coefficients; this yields the non-singularity of~$M$ and completes the rank identity.
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