A note on the proof of a theorem of Katz
Let \(f_i\in\mathbb F_q[X_1,\dots,X_n]\) be polynomials of degree \(d_i\), \(1\leq i\leq r\), where \(d_1\geq\cdots\geq d_r\geq 1\). Denote the set of zeros of \(f_i\) in \(\mathbb F_q^n\) by \(Z(f_i)\). \textit{N. M. Katz} [Am. J. Math. 93, 485--499 (1971; Zbl 0237.12012)] proved that \(q^{\lceil\frac {n-d_1-\cdots-d_r}{d_1}\rceil}\) divides \(|Z(f_1)\cap\cdots\cap Z(f_r)|\). A more elementary proof of this result was given by \textit{D. Wan} [Am. J. Math. 111, 1--8 (1989; Zbl 0683.12010)]. Here the author gives a new and simpler proof of this result.
scientific article; zbMATH DE number 2173081