We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison–Singer problem to prove improved paving estimates. Working with Anderson’s paving formulation of Kadison–Singer instead of Weaver’s vector balancing version, we show that the machinery of interlacing polynomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called “mixed determinants” that have been previously studied in a different context by Borcea and Brändén. This approach allows us to show that any projection with diagonal entries 1/2 can be 4 paved, yielding improvements over the best known current estimates of 12. This approach also allows us to show that any projection with diagonal entries strictly less than 1/4 can be 2 paved, matching recent results of Bownik, Casazza, Marcus and Speegle. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have several pleasing combinatorial properties.

中文翻译：

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混合行列式和 Kadison-Singer 问题

我们采用 Marcus、Spielman 和 Srivastava 在 Kadison-Singer 问题证明中的论点来证明改进的铺路估计。使用 Anderson 的 Kadison-Singer 铺路公式而不是 Weaver 的矢量平衡版本，我们表明 Marcus、Spielman 和 Srivastava 的交错多项式机制也适用于这种情况。结果证明，相关的预期特征多项式与 Borcea 和 Brändén 之前在不同背景下研究过的所谓“混合行列式”有关。这种方法使我们能够证明对角线条目 1/2 的任何投影都可以铺成 4，从而比目前最著名的估计值 12 有所改进。这种方法还允许我们表明对角线条目严格小于 1/4 的任何投影可以铺2，匹配 Bownik、Casazza、Marcus 和 Speegle 最近的结果。我们还将寻找最佳铺路估计的问题与矩阵特征多项式的自然单参数变形的根区间的边界联系起来，该矩阵的特征多项式具有几个令人愉悦的组合特性。