论文标题
$ \ Mathbb f_p $ - 缩减下的缩减范围的决定因素
Determinant of $\mathbb F_p$-hypergeometric solutions under ample reduction
论文作者
论文摘要
当高几何解决方案是一维积分时，我们考虑$ \ Mathbb c $上的Kz微分方程。我们还考虑了有限字段$ \ mathbb f_p $上的相同的微分方程。我们研究了这些微分方程的多项式解决方案，超过$ \ Mathbb f_p $，该方程在先前与v。\，Schechtman的工作接头中构建，并称为$ \ Mathbb f_p f_p $ -Hypheremetric solutions。 $ \ mathbb f_p $ -HyPerememetric解决方案的空间的尺寸取决于质量数$ P $。我们说，如果$ \ Mathbb f_p $ -Hypergeometric解决方案的空间的尺寸是最大的，即相当于$ \ $ \ thybb c $，则KZ方程对Prime $ p $的减少充分减少，也就是说。在大量降低的假设下，我们证明了基础$ \ MATHBB f_p $ - HYPHENEPENEMETRICTRITIC SOLUTION的矩阵矩阵的决定性公式。该公式类似于基础复杂高几何解决方案矩阵矩阵的相应公式，其中二项式$（z_i-Z_JJ）^{m_i+m_jj} $被替换为$（z_i-Z_jj）^{m_i+m_j+m_j+m_j+m_j+m_j+m_j+is $ $ $ $ $ $ $ $ $ $ $（x） $ \ mathbb f_p $ -analog $γ_ {\ mathbb f_p}（x）$在$ \ mathbb f_p $上定义。
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 dimension of the space of $\mathbb F_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $\mathbb C$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis $\mathbb F_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j)^{M_i+M_j}$ are replaced with $(z_i-z_j)^{M_i+M_j-p}$ and the Euler gamma function $Γ(x)$ is replaced with a suitable $\mathbb F_p$-analog $Γ_{\mathbb F_p}(x)$ defined on $\mathbb F_p$.
