Improvements upon the Chevalley-Warning-Ax-Katz-type estimates
Let \(f\) be a polynomial over a finite field \(\mathbb F_q\) with \(q\) elements and let \(N(f=0)\) denote the number of zeros of \(f\) in \(\mathbb F_q\). Let \(e_p\) denote the exponent of \(p\) in \(N(f=0)\). The Chevalley-Warning, Ax-Katz and Moreno-Moreno theorems give non-trivial lower bounds for \(e_p\) of increasing strength. Given \(f\) the authors define two associated polynomials \(\hat f\) and \(\skew6\hat{\hat f}\) that are possibly of lower degree than \(f\), but have the same number of zeros. On applying the above classical results with \(\hat f\) or \(\skew6\hat{\hat f}\) instead of \(f\), one then sometimes obtains a larger lower bound for \(e_p\). In case of \(\hat f\) the proof that \(N(f=0)=N({\hat f}=0)\) is nice, short and elementary. For \(\skew6\hat{\hat f}\) it relies on results from the earlier paper [Finite Fields Appl. 12, 681--692 (2006; Zbl 1169.11055)] by the same authors.
scientific article; zbMATH DE number 5124510