Authors: Juan Palencia, Tanner Probst, Clair Yeaman
Mentors: Machiel Van Frankenhuijsen
Insitution: Utah Valley University
Abstract: The ABC conjecture is an unsolved problem in mathematics first formulated by Joseph Oesterlé and David Masser. The conjecture attempts to show an analogy between polynomials and integers. This insight arises from the Stothers-Mason Theorem (also known as Mason’s Theorem). The theorem states that the maximum degree of a polynomial is always equal to or less than the number of distinct roots minus one. The ABC conjecture attempts to bridge this analogy by defining the height as the maximum algebraic number in the sum a + b = c where a, b and c are relatively prime. Masser and Oesterle define the radical as the sum of log p where p divides abc. Mason’s theorem states that the height is less than the radical minus one. It turns out that this is not true for the integers. Thus Masser and Oesterle formulated a conjecture which may be true. Over the past decades, this inequality has been strengthened and reformulated. Shinichi Mochizuki has recently published what he claims to be a proof of the conjecture, but many mathematicians are currently still working to understand it. The aim of our research is to better understand the height and radical in the ABC conjecture. In addition, we intend to explore the possibilities of expanding the Stothers-Mason Theorem. Since the ABC conjecture is concerned with an analogy between commutative rings (integers and polynomials), we are interested in seeing whether Mason’s Theorem for polynomials also holds for non-commutative rings, such as matrix rings and the ring of quaternions. In addition, we intend to explore the connection between the factorization of polynomials and finite Abelian groups.