A reduction for counting the number of zeros of general diagonal equation over finite fields
The authors provide an efficient reduction for counting the number of zeros of the so-called general diagonal equation over finite fields, which is of the form \[ a_1x_{11}^{k_{11}}\dots x_{1n_1}^{k_{1n_1}}+ \dots +a_rx_{r1}^{k_{r1}}\dots x_{rn_r}^{k_{rn_r}} = 0, \] where \(a_i \in \mathbb F_q^* ,\;k_{ij} \in {\mathbb Z}^+\) for \(1\leq i\leq r,\;1\leq j\leq n_{i}\). This generalizes the corresponding results for the simple diagonal equation of the form \(a_1x_1^{k_1}+ \dots + a_nx_n^{k_n} = 0 \) and the \(\ell\)-linear equation of the form \(a_1x_{11}\dots x_{1\ell}+\dots + a_rx_{r1}x_{r\ell} = 0.\) Furthermore, they show that the Chevalley-Warning-Ax-Katz type estimates can be improved in the cases of general diagonal equations by this reduction.
scientific article; zbMATH DE number 5503973