2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131670The 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.19 pages; Submitted to journal, currently under reviewCommutative AlgebraAlgebraic Geometry13P10; 15A63text