  The aim of this paper is to unveil an unexpected relationship between the
normal form of a polynomial with respect to a polynomial ideal and the more
geometric concept of orthogonality. We present a new way to calculate the
normal form of a polynomial with respect to a polynomial ideal I in the ring of
multivariate polynomials over a field K, provided the field K is finite and the
ideal I is a vanishing ideal. In order to use the concept of orthogonality, we
introduce a symmetric bilinear form on a vector space over a finite field.
