On indices and monogenity of certain classes' number fields defined by quadrinomials

Let $K$ be an algebraic number field of degree $n$ with ring of integers $\mathbb{Z}_K$.
The field $K$ is called monogenic if $\mathbb{Z}_K=\mathbb{Z}[\theta]$ for some primitive element $\theta$ of $Z_K$.
Equivalently, $(1,\theta,\ldots,\theta^{n-1})$ forms a $\mathbb{Z}$-basis of $\mathbb{Z}_K$.
Such a basis is called a power integral basis of $\mathbb{Z}_K$.

If the ring $\mathbb{Z}_K$ does not admit any power integral basis, then $K$ is said to be non-monogenic.
The monogeneity of number fields and the construction of power integral bases have been widely studied in both classical and recent research, and many interesting open problems related to these topics remain the subject of active investigation. One of the most important contributions to this problem was made by Gy\H{o}ry, who effectively proved that there exist only finitely many $\mathbb{Z}$-equivalence classes of elements $\theta$, generating a power integral basis of $\mathbb{Z}_K$.

In this talk, for some fixed positive integers $n$, $m$, and $s$, we study the prime common divisors of the indices associated with certain infinite families of number fields defined by quadrinomials of the form $x^n + a x^m + b x^s + c$.
As an application of our results, we provide some explicit conditions on $a$, $b$, and $c$ under which these number fields are not monogenic.
Our approach is based on Ore's classical theorem on the decomposition of primes in number fields, which is formulated using Newton polygon techniques.