The Index of Composition of polynomials and its applications
Abstract
The index of a given monic irreducible polynomial f(x)∈ℤ[x] having a root θ is the index of ℤ[θ] in the ring of algebraic integers of ℚ(θ). Further, f(x) is monogenic if its index is $1$. In this article, we prove that the index of f(x) always divides the index of f(g(x)), where f(x), g(x)∈ ℤ[x]. Additionally, we give the precise power of the index of f(x) dividing the index of f(g(x)). Also, we provide a necessary condition for the monogenity of f(g(x)). As an application of our results, we prove that the n-fold composition fn(x)=(f∘ f⋯∘ f)(x) of a polynomial f(x) is monogenic for all n∈ℕ if and only if the sequence ⟨ In⟩ is convergent, where In is the index of fn(x). Moreover, we show that for a given n∈ℕ, if In is mth power free integer, then the n-tower ℚ⊆ K1 ⊆⋯ ⊆ Kn−1 ⊆ Kn, defined by n iterates of a polynomial f(x)∈ℤ[x] contains at least max{⌊ n−log°fm ⌋,0} monogenic number fields.
Related
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- Reducible fibers of polynomials over integers: the Hilbert-Siegel problem over ℚ and its applications
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- Bounds on the realizations of zero-nonzero patterns and sign conditions of polynomials restricted to varieties and applications
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BibTeX
@misc{kumar2026index,
title = {{The Index of Composition of polynomials and its applications}},
author = {Surender Kumar and Sumandeep Kaur},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36877},
eprint = {2609.36877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36877},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Index of Composition of polynomials and its applications
AU - Surender Kumar
AU - Sumandeep Kaur
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36877
UR - https://arxiv.org/abs/2609.36877
AB - The index of a given monic irreducible polynomial f(x)∈ℤ[x] having a root θ is the index of ℤ[θ] in the ring of algebraic integers of ℚ(θ). Further, f(x) is monogenic if its index is $1$. In this article, we prove that the index of f(x) always divides the index of f(g(x)), where f(x), g(x)∈ ℤ[x]. Additionally, we give the precise power of the index of f(x) dividing the index of f(g(x)). Also, we provide a necessary condition for the monogenity of f(g(x)). As an application of our results, we prove that the n-fold composition f^n(x)=(f∘ f⋯∘ f)(x) of a polynomial f(x) is monogenic for all n∈ℕ if and only if the sequence ⟨ I_n⟩ is convergent, where I_n is the index of f^n(x). Moreover, we show that for a given n∈ℕ, if I_n is m^th power free integer, then the n-tower ℚ⊆ K_1 ⊆⋯ ⊆ K_n−1 ⊆ K_n, defined by n iterates of a polynomial f(x)∈ℤ[x] contains at least max{⌊ n−log_°fm ⌋,0} monogenic number fields.
ER -