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 a(1)x(11)(k11...)x(ln1)(kln) +(...)+a(r)x(r1)(kln1k...)x(rnr)(krnr) = 0. This generalizes the corresponding results for the simple diagonal equation of the form a(1)x(1)(kl)+(...)+a(n)x(n)(kn)=0 and the l-linear equation of the form a(1)x(11)(...)x(1l)+(...)+a(r)x(r1)(...)x(rl)=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. (c) 2005 Elsevier Inc. All rights reserved.