Varchenko introduced a distance function on chambers of hyperplane arrangements that he called quantum bilinear form. That gave rise to a determinant indexed by chambers whose entry in position ( C , D ) is the distance between C and D : that is the Varchenko determinant. He showed that that determinant has a nice factorization. Later, Aguiar and Mahajan defined a generalization of the quantum bilinear form, and computed the Varchenko determinant given rise by that generalization for central hyperplane arrangements and their cones. This article takes inspiration from their proof strategy to compute the Varchenko determinant given rise by their distance function for apartment of hyperplane arrangements. Those latter are in fact realizable conditional oriented matroids.

中文翻译：

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公寓的 Varchenko 行列式

Varchenko 在超平面排列的腔室中引入了一个距离函数，他称之为量子双线性形式。这产生了一个由腔室索引的行列式，其位置（ C ， D ）是 C 和 D 之间的距离：这就是 Varchenko 行列式。他表明该行列式有一个很好的因式分解。后来，Aguiar 和 Mahajan 定义了量子双线性形式的推广，并计算了由中心超平面排列及其锥体的推广引起的 Varchenko 行列式。本文从他们的证明策略中汲取灵感，计算由他们的超平面排列公寓的距离函数引起的 Varchenko 行列式。后者实际上是可实现的条件定向拟阵。