Monogeneity of iterated polynomial compositions

Determine the conditions under which the composition $(fcirc g^N)(x)$ is monogenic for monic polynomials f(x) and g(x) in Z[x] and an iteration count N.

Background

The paper turns from one-step compositions to dynamical towers generated by repeated composition. It establishes criteria for compositions fcirc HN when H is derived from a trinomial of the form h(x)=xm+a x{m-1}+b, and it gives further specialized results for binomial iterates.

The general question for arbitrary monic f and g remains broader than the theorem proved: the paper’s characterization depends on the special structure of the inner polynomial and on irreducibility and valuation hypotheses. Thus the general monogeneity problem for iterated compositions is left unresolved outside those structured families.

References

Having understood the monogeneity of polynomial compositions, a natural next step in studying number fields of higher degree is to consider iterates of a polynomial and ask whether the resulting iterated polynomials remain monogenic. These observations leads to the following natural question concerning the monogeneity of polynomial iterates: Let $f(x)$ and $g(x)$ be monic polynomials in $\mathbb{Z}[x]$. Under what conditions is $(f\circ gN)(x)$ 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 2.6 (page number unavailable)