A Local Approach to Monogenity with an Application to Lenny Jones'Conjecture

Abstract

The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion for the $p$-maximality of orders generated by roots of arbitrary monic irreducible polynomials. As an application, we apply this to irreducible polynomials of the type $f(X)=X^n+A(BX+1)^m,$ where $1\le m<n$, $\gcd(n,mB)=1$, and $A,B\in\mathbb{Z}\setminus\{0\}$. We show that $f$ is monogenic if and only if both $A~\text{and}~n^n+(-1)^{n+m}B^n(n-m)^{\,n-m}m^mA$ are square-free. This provides a new proof of the main theorem of \cite{KK}, thereby proving Lenny Jones'conjecture \cite[Conjecture 4.1]{LJ}. Furthermore, we obtain explicit infinite families of irreducible non-monogenic polynomials, including trinomial, quadrinomial, and power-compositional families.