The Ideal of Vanishing Polynomials and the Ring of Polynomial Functions

TL;DR

This paper investigates the structure of vanishing polynomials over various rings, including integers modulo n and general commutative rings, providing new insights into their ideals and polynomial functions.

Contribution

It introduces new results on generating vanishing polynomials and establishes isomorphisms between vanishing polynomials of composite rings and their components.

Findings

Characterization of vanishing polynomials over al_n

Isomorphism between vanishing polynomials of rings and their components

Generalization to arbitrary commutative rings

Abstract

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial functions. In particular, we provide several results on generating vanishing polynomials. We first analyze the ideal of vanishing polynomial over $\mathbb{Z}_n$, the ring of the integers modulo $n$. We then establish an isomorphism between the vanishing polynomials of a ring and the vanishing polynomials of the constituent rings in its decomposition. Lastly, we generalize our results to study the ideal of vanishing polynomials over arbitrary commutative rings.