# Results on Vanishing Polynomials and Polynomial Root Counting

## TL;DR

This paper investigates vanishing polynomials over commutative rings, determining minimal degrees, classifying ideals, and analyzing root restrictions, with implications for ring theory and applications in mathematics and engineering.

## Contribution

It provides new results on minimal degrees of vanishing polynomials, classifies their ideals over specific rings, and introduces techniques to limit roots over finite rings.

## Key findings

- Minimum degree of monic vanishing polynomials determined for certain rings
- Partial classification of vanishing polynomial ideals over prime and prime square order rings
- A technique to restrict the number of roots of polynomials over finite rings

## Abstract

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close connections to both ring theory and the technical applications of polynomials, along with numerous applications to other mathematical and engineering fields. We first determine the minimum degree of monic vanishing polynomials over a specific infinite family of rings of a specific form and consider a generalization of the notion of a monic vanishing polynomial over a subring. We then present a partial classification of the ideal of vanishing polynomials over general commutative rings with identity of prime and prime square orders. Finally, we prove some results on rings that have a finite number of roots and propose a technique that can be utilized to restrict the number of roots polynomials can have over certain finite commutative rings.