On the monogenity of composed polynomials

Let g(x)∈Z[x] be a monic irreducible polynomial of degree n. We say that g(x) is monogenic if, for a root θ of g(x), the set {1,θ,…,θn−1} forms an integral basis of the ring of integers ZK of the number field K=Q(θ). Consider f(x)=xn+ax2+dx+b∈Z[x] with d2=4ab, and g(x)=xm+c∈Z[x], where n≥3 and m≥1, such that (f°g)(x)=(xm+c)n+a(xm+c)2+d(xm+c)+b is irreducible over Q. In this study, we establish necessary and sufficient conditions involving a, b, c, d, m, n for the polynomial (f°g)(x) to be monogenic. Additionally, we examine the nature of solutions to specific differential equations, and present a class of monogenic polynomials with non-square-free discriminants as an application.