Infinite families of monogenic quadrinomials, quintinomials and sextinomials

Let $f(x)\in \mathbb Z [x]$ be monic, with $\deg (f)=n$. We say $f(x)$ is \emph {monogenic} if $f(x)$ is irreducible over $\mathbb Q $ and $\{1,\alpha ,\alpha ^2,\ldots , \alpha ^{n-1}\}$ is a basis for the ring of integers of $K=\mathbb Q (\alpha )$, where $f(\alpha )=0$. In this article, we derive a new polynomial discriminant formula, and we use it to construct infinite families of monogenic quadrinomials, quintinomials and sextinomials for any degree $n\ge 3,4,5$, respectively. These results extend previous work of the author. We also give a brief discussion concerning the adaptation of our approach beyond sextinomials.