A note on the proof of a theorem of Katz

Let fi∈Fq[X1,…,Xn] be polynomials of degree di, 1⩽i⩽r, where d1⩾⋯⩾dr⩾1. Denote the set of zeros of fi in Fqn by Z(fi). Katz proved that q⌈n-d1-⋯-drd1⌉ divides |Z(f1)∩⋯∩Z(fr)|. A more elementary proof of this result was given by Wan. We found a new and much simpler proof of this result.