A note on the proof of a theorem of Katz

摘要

Let f_i ∈ F_q[X_1, ..., X_n] be polynomials of degree d_i 1 ≤ i ≤ r where d_1 ≥ ... ≥ d_r ≥ 1. Denote the set of zeros of f_i in F_q~n by Z(f_i). Katz proved that q~([(n-d_1-...-d_r)/d_1]) divides |Z(f_1) ∩ ... ∩ Z(f_r)|. A more elementary proof of this result was given by Wan. We found a new and much simpler proof of this result.