Simultaneous monogeneity of a polynomial and its composition

Determine the conditions under which two monic polynomials f(x) and g(x) in Z[x] are such that both f(x) and the composition f(g(x)) are monogenic.

Background

The paper distinguishes monogeneity of a number field from monogeneity of a defining polynomial and investigates how monogeneity behaves under polynomial composition. It proves necessary and sufficient criteria for compositions with a specific trinomial inner polynomial h(x)=xm+a x{m-1}+b, as well as related binomial and iterated cases.

The stated problem is broader than these results because it concerns arbitrary monic f(x),g(x) in Z[x]. The paper notes that arithmetic invariants of composed polynomials do not behave straightforwardly, making a general characterization of simultaneous monogeneity unresolved beyond the families treated in the paper.

References

This also leads to a natural question concerning monogeneity: Let $f(x)$ and $g(x)$ be monic polynomials in $\mathbb{Z}[x]$. Under what conditions are both $f(x)$ and $f(g(x))$ monogenic?

— On The Index of Polynomial Compositions over Valued Fields  (2610.02111 - Jakhar et al., 1 Oct 2026) in Section 1, Introduction, immediately before Theorem 2.1 (page number unavailable)