Based on results obtained during his 2018 REU, Associate Professor of Mathematics Chad Awtrey has published a research article in the most recent issue of Involve, a Journal of Mathematics.
Associate Professor of Mathematics Chad Awtrey has published a research article in the most recent issue of Involve, a Journal of Mathematics based on results obtained during his 2018 research experiences for undergraduate program.
The paper — “The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of p-adic fields“, Involve, a journal of mathematics, 13, no. 5, 747-758 (2020) — was coauthored with several students and faculty from Awtrey’s 2018 summer REU (research experiences for undergraduate) program, funded by the National Security Agency. The authors explored properties of the discriminants of polynomials with p-adic coefficients, where p is a prime number. The discriminant of a polynomial is a special number, computed using the polynomial’s coefficients, that gives some arithmetic information about the polynomial and its roots. For example, the discriminant of the quadratic polynomial ax^2+bx+c is b^2-4ac; this number is 0 if and only if the quadratic polynomial has a repeated root.
By studying the base p representations of the discriminants, the authors were able to show that, under certain divisibility conditions, the first nonzero digit of the base p representation of a polynomial’s discriminant is a defining characteristic of that polynomial. In other words, if two polynomials of the same degree have roots that can be expressed algebraically in terms of each other’s roots, then the first nonzero digit of the base p representations of their respective discriminants are the same. As one application, they show this particular digit completely identifies the symmetry properties in the case the polynomial has degree p, thereby giving an elementary method for confirming previous research in this area.
The authors include: faculty members Sebastian Pauli (UNC-Greensboro) and Scott Zinzer (Aurora University); graduate student Sandi Rudzinski (UNC-Greensboro), and undergraduates Alex Gaura (Princeton University) and Ariel Uy (Carnegie Mellon University).