AI 中文总结
该研究在$\boldsymbol{\rm \bf C}^6$中构造19个高斯整数项向量实现相位恢复，将6维最小相位恢复测量数的上界降至19，结合已知下界得其为18或19，且通过精确有理计算完成认证。
AI 中文摘要
我们展示了$\boldsymbol{\rm \bf C}^6$中19个具有高斯整数项的显式向量，它们可用于相位恢复：强度$|\boldsymbol{\rm \bf \text{⟨}}a_j,x\boldsymbol{\rm \bf \text{⟩}}|^2$（$j=1,\boldsymbol{\rm \bf \text{…}},19$）能确定$\boldsymbol{\rm \bf C}^6$中的每个$x$，仅相差一个模为1的标量。这给出$m_{\boldsymbol{\rm \bf C}}(6)\boldsymbol{\rm \bf ≤}19=4d-5$，比通用计数$4d-4$少1；结合Wang和Xu给出的下界$m_{\boldsymbol{\rm \bf C}}(6)\boldsymbol{\rm \bf ≥}18$，6维空间中的最小测量数为18或19。该结论通过精确的有理计算验证，且可归档并重复运行：经基础约化后，相位恢复等价于5个5元二次方程构成的方程组的实零点集为空，该方程组的归档消元式是一个无实根的14次无平方因子整系数多项式。
英文摘要
We exhibit nineteen explicit vectors in $\mathbb{C}^6$, with Gaussian-integer entries, that do phase retrieval: the intensities $|\langle a_j,x\rangle|^2$, $j=1,\dots,19$, determine every $x\in\mathbb{C}^6$ up to a unimodular scalar. This gives $m{\mathbb{C}}(6)\le 19=4d-5$, one below the generic count $4d-4$; combined with the lower bound $m{\mathbb{C}}(6)\ge 18$ of Wang and Xu, the minimal measurement number in dimension six is 18 or 19. The claim is verified by an exact rational computation, archived and rerunnable: after an elementary reduction, phase retrieval is equivalent to the emptiness of the real zero set of a system of five quadratic equations in five unknowns, and the archived eliminant of that system is a squarefree integer polynomial of degree fourteen with no real roots.
