  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.
