A reduction for counting the number of zeros of general diagonal equation over finite fields

We 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 . This generalizes the corresponding results for the simple diagonal equation of the form and the l-linear equation of the form a1x11 x1l+ +arxr1 xrl=0. Furthermore, we show that the Chevalley–Warning–Ax–Katz type estimates can be improved in the cases of general diagonal equations by our reduction.