AI 中文总结
本文将Solymosi和Zahl关于一维等距仿射平面族有限性的结果推广至任意维数，通过秩下降等方法证明了等距仿射k-平面族基数上确界的双指数显式上界。
AI 中文摘要
设0≤k<n，记𝒩(k,n)为ℝⁿ中两两欧氏距离均等于1的仿射k-平面有限族的基数的上确界。对于k=1，Solymosi和Zahl近期已证明𝒩(1,n)的有限性。我们将该结果推广至任意维数的仿射子空间，并证明了完全显式的界：𝒩(k,n)≤2^(2^(10(k+1)(n−k)))。高维中的主要困难在于，有理距离公式会沿dim(U+V)的多个可能秩层退化，而非仅沿平行轨迹退化。对每个固定秩r，我们用涉及所有子式的全局平方和表达式替代非零Gram子式的选择，得到多项式关系Hᵣ(x,y)=0，其既编码单位距离方程和高秩条件，也在对角线上消失。我们通过受限扎里斯基闭包传播该关系，并将所得代数集分解为显式有界数量的连通纳什子流形。解析对角近似论证表明，在每个纳什块内，dim(Uₓ+Uᵧ)的最大可能值必然下降。对所有可能秩迭代该秩下降过程，并结合有限多个标准仿射格拉斯曼图，即可得到所述双指数上界。
英文摘要
Let \(0\leqslant k<n\), and let \(\mathcal N(k,n)\) denote the supremum of the cardinalities of finite families of affine \(k\)-planes in \(\R^n\) whose pairwise Euclidean separation distances are all equal to one. For \(k=1\), the finiteness of \(\mathcal N(1,n)\) was recently established by Solymosi and Zahl. We extend this result to affine subspaces of arbitrary dimension and prove the fully explicit bound \[ \mathcal N(k,n) \leqslant 2^{\,2^{\,10(k+1)(n-k)}}. \] The main difficulty in higher dimensions is that the rational distance formula degenerates along several possible rank strata of \(\dim(U+V)\), rather than only along the parallel locus. For each fixed rank \(r\), we replace the choice of a nonvanishing Gram minor by a global sum-of-squares expression involving all minors. This produces a polynomial relation \(H_r(x,y)=0\) which encodes both the unit-distance equation and an upper-rank condition, while also vanishing on the diagonal. We propagate this relation by means of the restricted Zariski closure and decompose the resulting algebraic set into an explicitly bounded number of connected Nash submanifolds. An analytic diagonal approximation argument shows that, within each Nash piece, the maximal possible value of \(\dim(U_x+U_y)\) must decrease. Iterating this rank descent over all possible ranks and combining the finitely many standard affine Grassmann charts yields the stated double-exponential upper bound.
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