On The Index of Polynomial Compositions over Valued Fields
Abstract
Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions f(h(x)), where f(x) and h(x) are monic polynomials over an arbitrary Krull valuation ring and h(x) is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when f(h(x)) generates a monogenic field. In particular, we relate the monogeneity of the composition to that of f(x) and to explicit square-free conditions on the critical values of h(x). We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.
Related
- On the index of generalized trinomials over Valued fields
- Monogenic Fields from Polynomial Compositions with Applications
- Abeloid varieties over higher rank valued fields: Raynaud uniformization and Néron models
- Improved computation of polynomial roots over number fields when using complex embeddings
- Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index
BibTeX
@misc{jakhar2026index,
title = {{On The Index of Polynomial Compositions over Valued Fields}},
author = {Anuj Jakhar and Ravi Kalwaniya and Shanta Laishram and Prabhakar Yadav},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02111},
eprint = {2610.02111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02111},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On The Index of Polynomial Compositions over Valued Fields
AU - Anuj Jakhar
AU - Ravi Kalwaniya
AU - Shanta Laishram
AU - Prabhakar Yadav
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02111
UR - https://arxiv.org/abs/2610.02111
AB - Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions f(h(x)), where f(x) and h(x) are monic polynomials over an arbitrary Krull valuation ring and h(x) is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when f(h(x)) generates a monogenic field. In particular, we relate the monogeneity of the composition to that of f(x) and to explicit square-free conditions on the critical values of h(x). We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.
ER -