Canonical representatives for residue classes of a polynomial ideal and orthogonality

TL;DR

This paper explores a novel connection between polynomial normal forms relative to ideals and orthogonality, introducing a new computational method for ideals over finite fields using a symmetric bilinear form.

Contribution

It presents a new approach to compute polynomial normal forms via orthogonality, linking algebraic and geometric concepts in finite field polynomial ideals.

Findings

Established a symmetric bilinear form for finite fields

Developed a new method for normal form calculation

Linked algebraic and geometric perspectives

Abstract

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.