论文标题
$ \ mathbb {f} _q $ - 对角超曲面和超几何功能的点
Number of $\mathbb{F}_q$-points on Diagonal hypersurfaces and hypergeometric function
论文作者
论文摘要
令$d_λ^d $表示有限字段$ \ mathbb {f} _q $ \ begin \ begin {align*}在有限场上的对角线横向表面的单次变形家庭 d_λ^d：x_1^d+x_2^d+\ cdots+x_n^d =λdx_1^{h_1} X_2^{ h_i = d $，$ \ gcd（d，h_1，h_2，\ ldots，h_n）= 1 $。当$ d = n $，即，$ h_1 = h_1 = \ cdots = h_n = 1 $时，dwork hypersurface就是这种情况。 $ \ mathbb {f} _q $ - dwork hypersurfaces上的$ \ mathbb {f} _q $点的公式是已知的。在本文中，我们为$d_λ^d $上的$ \ mathbb {f} _q $ - 点的数量提供了一个公式。
Let $D_λ^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$, $h_i\geq1$, $\sum_{i=1}^n h_i=d$, and $\gcd(d,h_1,h_2,\ldots, h_n)=1$. The Dwork hypersurface is the case when $d=n$, that is, $h_1=h_2=\cdots =h_n=1$. Formulas for the number of $\mathbb{F}_q$-points on the Dwork hypersurfaces in terms of McCarthy's $p$-adic hypergeometric functions are known. In this article we provide a formula for the number of $\mathbb{F}_q$-points on $D_λ^d$ in terms of McCarthy's $p$-adic hypergeometric function which holds for $d\geq n$.
