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We study the set of algebraic objects known as vanishing polynomials (the setof polynomials that annihilate all elements of a ring) over general commutativerings with identity. These objects are of special interest due to their closeconnections 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 aspecific infinite family of rings of a specific form and consider ageneralization of the notion of a monic vanishing polynomial over a subring. Wethen present a partial classification of the ideal of vanishing polynomialsover 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 andpropose a technique that can be utilized to restrict the number of rootspolynomials can have over certain finite commutative rings.

Results on Vanishing Polynomials and Polynomial Root Counting