Nonsingular zeros of polynomials defined over finite fields

Abstract

The aim of this paper is to study the existence of nontrivial, nonsingular zeros of a nonhomogeneous polynomial defined over a finite field. To accomplish this, we determine conditions that guarantee the existence of a prescribed number of nonsingular zeros of a homogeneous form f over a finite field k that are not zeros of a homogeneous form h when f, h are relatively prime. The cases of quadratic and cubic polynomials are considered in detail. This extends previous results that have usually considered only the homogeneous case.