通过对角GP图研究对角方程组的解的数量：一般情形与埃尔米特形式情形
On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases
- Universidad Nacional de Córdoba（科尔多瓦国立大学）
- CONICET（阿根廷国家科学研究委员会）
- FaMAF（数学与物理系）
- CIEM（数学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过引入对角GP图，给出首一对角方程组解的数量的三种表达式，并结合埃尔米特形式图的谱推导了特定同次对角方程组解的数量的组合公式与显式结果。
AI中文摘要：
对于任意的正整数m、s，我们研究首一对角方程组的解的数量$N_{m\times s,q}(\kappa, \beta)$，其中解为$(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$，该方程组形式为$X_{1}^{k_i} + \cdots + X_{s}^{k_i}= \beta_i$（$1\le i \le m$），这里$\kappa=(k_1,\ldots,k_m) \in \mathbb{N}^m$，$\beta=(\beta_1,\ldots,\beta_m) \in (\mathbb{F}_q)^m$。我们证明该解的数量可通过我们在此引入的新一类图——对角GP图$\Gamma(\kappa,q)$的相关数据得到，这类图是凯莱图，形式为$\Gamma(\kappa,q) = Cay(\mathbb{F}_{q}^{m}, R_{\kappa})$，其中$R_{\kappa} = \{ (x^{k_1},\ldots,x^{k_m}): x \in \mathbb{F}_{q}^*\}$，$\kappa=(k_1,\ldots,k_m)\in \mathbb{N}^{m}$。特别地，我们给出$N_{m\times s,q}(\kappa, \beta)$的三种不同表达式：一种基于图的行走，一种基于$\Gamma(\kappa,q)$的邻接矩阵，最后一种基于$\Gamma(\kappa,q)$的谱。最后，我们通过埃尔米特形式图（可视为对角GP图）的已知谱，明确推导首一同次对角方程组的解的数量$N_{m}(s,q) = N_{m\times s,q}(\kappa_\ell, 0)$的组合公式，该方程组形式为$X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0$（$1\le i \le m$），其中$\kappa_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$且$m\ge 2$。对于任意正整数m、s，我们给出$N_m(s,q) \in \mathbb{Z}[q]$的一般求和公式与递推公式；对于$N_{1}(s,q)$、$N_{2}(s,q)$以及$1\le s \le 5$时的$N_{m}(s,q)$这些小情形，我们给出显式表达式。
英文摘要：
For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(κ, β)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= β_i, \qquad (1\le i \le m), $$ with $κ=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $β=(β_1,\ldots,β_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of \textit{diagonal} GP-graphs $Γ(κ,q)$. This is a new family of graphs that we introduce here, $Γ(κ,q)$, with $κ= (k_1,\ldots,k_m) \in \mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \in R_κ = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}$. In particular, we give three different expressions for $N_{m\times s,q}(κ, β)$: one in terms of walks, another in terms of adjacency matrices of $Γ(κ,q)$ and the last one in terms of the spectrum of $Γ(κ,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(κ_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $κ_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.
