Monogenic polynomials and symmetric Galois groups: a quantitative study

The main purpose of this article is to extract infinite families of polynomials of the form $$ t^n + c(at^k + b)^m \in \mathbb {Z}[t] $$ , where $$ k \ge 2 $$ and the degree $$ n > mk + 1 $$ is fixed, such that the discriminants are not necessarily squarefree, yet the polynomials are either monogenic or possess the full symmetric group as their Galois group. In particular, we establish lower bounds on the number of monogenic polynomials of this form for the cases $$(i)$$ $$ m = 1 $$ , and $$(ii)$$ $$ b = 1 $$ when $$ m \ge 2 $$ . Furthermore, for any prime number $$ p $$ , we provide a lower bound on the number of monogenic polynomials of the form $$ t^p + c(at^k + b)^m \in \mathbb {Z}[t] $$ of degree $$ p $$ , whose Galois group over $$ \mathbb {Q} $$ is isomorphic to the symmetric group $$ S_p $$ .