On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases

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,\ld