排列及其对偶的超几何周期矩阵的行列式

Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants

The n-dimensional hypergeometric integrals associated with a hypersphere arrangement are formulated by the pairing of n-dimensional twisted cohomology and its dual. Under the condition of general position there are stated some results which concern an explicit representation of the standard form by a special (NBC) basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms written by Cayley-Menger minor determinants. Gauss-Manin connection is also formulated and is explicitly presented in two simplest cases.

由超平面排列引起的超几何函数的单值性

Monodromy of hypergeometric functions arising from arrangements of hyperplanes

给定$¶^n$中的超平面的一种排列，可能有非正规交叉，我们给出了沿我们的排列具有对数极点的$q$ -形式层的上同调的消失引理。给出了Orlik-Solomon代数的理想关系$\cal J$的一个基.在一定的属性条件下，H.~埃斯诺，V.~ Schechtman和E.~认为局部系统的上同调是由Aomoto复形给出的。我们将这一结果推广到通过变形我们的安排获得的局部系统的变形。我们计算了这种情况下的Gaussian-Manin联系。我们给出了Gaussian-Manin丛的一个基，借助于$\cal J$的基，我们给出了计算这种联络的一个表示的方法。从这里，与K-T的结果。陈或P。~德利涅，我们可以计算单值表示。这给出了超几何函数的一个推广。

超几何周期矩阵的行列式

The Determinant of a Hypergeometric Period Matrix

考虑实仿射空间上的函数$U=e^{-f_0}\prod_j^N f_j^{α_j}$，这里$f_0,..,f_N$是线性函数，$α_1, ...,α_N$是复数.函数$f_1, ..., f_N$的零点在仿射空间中形成超平面的排列。我们研究的周期矩阵的超几何积分与安排和功能$U$和计算其行列式作为交替产品的伽玛功能和临界点的功能$f_0,..., f_N$的安排。在最简单的例子$N=1, f_0=f_1=t$中，行列式公式的形式为$\int_0^\infty e^{-t} t^{α-1} dt=Γ(α).$我们还给出了Selberg型指数积分的行列式公式.在这种情况下，超平面的排列是特殊的，并承认一个对称群，周期矩阵被分解成块对应于不同的表示的对称群的空间上的超几何积分与安排。我们计算对应于平凡表示的块的行列式。

On the rank of an $A$-hypergeometric $D$-module versus the normalized volume of $A$

The rank of an $A$-hypergeometric $D$-module $M_A(β)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $β\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $β$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $β\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(β))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $β$ such that this ratio is at least $2$ is an affi…

Determinant of $\mathbb F_p$-hypergeometric solutions under ample reduction

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work joint with V.\,Schechtman and called the $\mathbb F_p$-hypergeometric solutions. The dimension of the space of $\mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimen…

Intersection numbers of twisted cycles and cocycles for degenerate arrangements

We study the intersection numbers defined on twisted homology or cohomology groups that are associated with hypergeometric integrals corresponding to degenerate hyperplane arrangements in the projective $k$-space. We present formulas to evaluate the intersection numbers in the case when exactly one $(k+1)$-tuple of the hyperplanes intersects at a point. As an application, we discuss the contiguity relations of hypergeometric functions in terms of the intersection numbers on twisted cohomology groups.