Generating Infinite Families of Monogenic Polynomials Using a New Discriminant Formula

Abstract

Recently, Otake and Shaska have given a formula for the discriminant of quadrinomials of the form $f(x) = x^n + t(x^2 + ax + b)$. In their concluding remarks, they ask if a formula can be found for the discriminant of $f(x) = x^n + tg(x)$ when $n > \deg(g) = 3$. Assuming that $f(x) = x^n + tg(x)$ is irreducible, and under certain restrictions on a polynomial related to $g(x)$, in this article we give a formula for the discriminant of $f(x)$, regardless of $\deg(g) \ge 1$. We then use our discriminant formula to generate some new infinite families of monogenic polynomials $f(x) = x^n + tg(x)$ with $n > \deg(g)$, when $g(x)$ is monic and $\deg(g)\in 2 \{2, 3, 4\}$.