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arxiv: 2606.09739 · v1 · pith:WGVVAHQLnew · submitted 2026-06-08 · 🧮 math.NT

Power Integral Bases in Polynomial Compositions

Aakash Choudhary , Supriya Pisolkar , Prabhakar Yadav This is my paper

Pith reviewed 2026-06-27 14:47 UTC · model grok-4.3

classification 🧮 math.NT
keywords monogenic polynomialspolynomial compositionsintegral basesnumber fieldsirreducibilitydiscriminantsdifferential equations
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The pith

Necessary and sufficient conditions on a, b, c, d, m, n determine when the composed polynomial is monogenic under irreducibility over Q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines monogeneity for polynomials of the special composed form (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b where the inner polynomial f satisfies a^2 = 4d. Assuming the full composition is irreducible over the rationals, it derives exact conditions on the six parameters that make the powers of a root form an integral basis for the number field. A sympathetic reader would care because monogenic fields simplify the ring of integers to a power basis, which streamlines arithmetic and discriminant calculations. The work also supplies a lower bound on the count of such monogenic examples and applies the conditions to related differential equations and to families with non-square-free discriminants.

Core claim

Assuming (f ∘ g)(x) is irreducible over Q, the polynomial is monogenic precisely when the parameters a, b, c, d, m, n satisfy the stated necessary and sufficient conditions; in that case the set {1, θ, …, θ^{mn−1}} is an integral basis of Q(θ) for any root θ.

What carries the argument

The composed polynomial (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b with the auxiliary relation a^2 = 4d on f, together with the parameter conditions that force the index of Z[θ] to be 1.

If this is right

  • When the parameter conditions hold, {1, θ, …, θ^{mn−1}} forms an integral basis of Q(θ).
  • The set of such monogenic composed polynomials has a positive lower bound on its cardinality.
  • Solutions to the associated differential equations display the behaviors derived from the monogeneity conditions.
  • The results produce explicit infinite families of polynomials whose discriminants are not square-free.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same composition technique might generate further infinite families once the a^2 = 4d restriction is relaxed.
  • The lower bound on monogenic examples supplies a concrete source of number fields for testing conjectures on class numbers or unit groups.
  • The differential-equation application suggests a possible bridge between monogenic orders and certain integrable systems.

Load-bearing premise

The composed polynomial must be irreducible over the rationals before the monogeneity conditions can be stated.

What would settle it

A concrete tuple of integers a, b, c, d, m, n that meets every listed condition yet for which the discriminant of the polynomial is strictly larger than the field discriminant of Q(θ) would falsify the claim.

read the original abstract

In this paper, we study the monogeneity of a special class of composed polynomials of the form $ (f \circ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b,$ where \( f(x) = x^n + a x^{n-1} + d x^{n-2} + b \in \mathbb{Z}[x] \) satisfies \( a^2 = 4d \) and \( g(x) = x^m + c \in \mathbb{Z}[x] \). Assuming that \( (f \circ g)(x) \) is irreducible over \( \mathbb{Q} \), we obtain necessary and sufficient conditions on the parameters \( a, b, c, d, m, n \) for the polynomial to be monogenic. These conditions help to identify when the set \( \{1, \theta, \dots, \theta^{mn-1}\} \) forms an integral basis of the number field \( \mathbb{Q}(\theta) \), where \( \theta \) is a root of \( (f \circ g)(x) \). We also provide lower bound for the counting of such monogenic polynomials. Furthermore, we study the behaviour of solutions to certain related differential equations and present a class of polynomials with non-square-free discriminants as an application of the main results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript examines monogeneity for the composed polynomials (f ∘ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b where f satisfies a² = 4d. Assuming irreducibility of (f ∘ g) over Q, it derives necessary and sufficient conditions on the parameters a, b, c, d, m, n such that the power basis {1, θ, …, θ^{mn-1}} is an integral basis of Q(θ). The paper also supplies a lower bound on the number of such monogenic polynomials and discusses applications to solutions of related differential equations together with examples of polynomials having non-square-free discriminants.

Significance. If the stated conditions are correctly derived, the work supplies explicit, usable criteria for monogeneity within a parametrized family of composed polynomials by exploiting the relation a² = 4d to control the index [O_K : Z[θ]]. The quantitative lower bound on the count of monogenic examples and the indicated applications to differential equations and non-square-free discriminants broaden the utility of the results within algebraic number theory.

minor comments (3)
  1. The abstract states that a lower bound is provided for the counting of monogenic polynomials but does not indicate the form of the bound, the range of parameters over which it applies, or the method used to obtain it.
  2. The connection between the main monogeneity theorems and the study of solutions to the related differential equations is mentioned only briefly; a short clarifying paragraph would help readers see how the two parts fit together.
  3. Notation for the composed polynomial and the parameters is introduced in the abstract; ensure that the same symbols are used consistently from the first page of the introduction onward.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper assumes irreducibility of (f ∘ g)(x) over Q as a prerequisite, then derives necessary and sufficient conditions on the parameters a, b, c, d, m, n (with the given relation a² = 4d) for monogeneity of the composed polynomial. This is a standard conditional derivation in algebraic number theory: the conditions are obtained from the structure of the order Z[θ] and its index in the ring of integers, without reducing the claimed result to a fitted parameter, self-definition, or load-bearing self-citation. No step equates a derived quantity to its input by construction, and the counting lower bound and differential equation applications are presented as consequences rather than tautologies. The derivation chain remains independent of its target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard domain assumption of irreducibility over Q together with the coefficient relation a² = 4d; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The composed polynomial (f ∘ g)(x) is irreducible over Q
    Explicitly stated in the abstract as the hypothesis under which the monogeneity conditions hold.

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Reference graph

Works this paper leans on

24 extracted references

  1. [1]

    Barman, A

    R. Barman, A. Narode, and V. Wagh. On the monogenity of polynomials with non-squarefree dis- criminants.arXiv preprint arXiv:2506.16496, 2025

    arXiv 2025

  2. [2]

    Cohen.A course in computational algebraic number theory, volume 138

    H. Cohen.A course in computational algebraic number theory, volume 138. Springer Science & Business Media, 2013

    2013

  3. [3]

    Cullinan

    J. Cullinan. https://studylib.net/doc/8187082/the-discriminant-of-a-composition-of-two.Depart- ment of Mathematics, Bard College, Annandale-On-Hudson, NY 12504

    arXiv

  4. [4]

    Dedekind

    R. Dedekind. ¨Uber den zusammenhang zwischen der theorie der ideale und der theorie der h¨ oheren congruenzen. 1878

  5. [5]

    P. Erd¨ os. Arithmetical properties of polynomials.J. London Math. Soc., 28:416–425, 1953

    1953

  6. [6]

    I. Ga´ al. On the monogenity of certain binomial compositions.JP Journal of Algebra, Number Theory and Applications, 57:1–16, 2022

    2022

  7. [7]

    I. Ga´ al. Monogenity and power integral bases: recent developments.Axioms, 13(7):429, 2024. POWER INTEGRAL BASES IN POLYNOMIAL COMPOSITIONS 15

    2024

  8. [8]

    I. Ga´ al. Diophantine equations and power integral bases: Theory and algorithms, 2nd edi. Birkh¨ aauser/Springer, Cham (2019)

    2019

  9. [9]

    Granville

    A. Granville. Abc allows us to count squarefrees.IMRN: International Mathematics Research Notices, 1998(19), 1998

    1998

  10. [10]

    Harrington and L

    J. Harrington and L. Jones. Monogenic binomial compositions.Taiwanese Journal of Mathematics, 24(5):1073–1090, 2020

    2020

  11. [11]

    Harrington and L

    J. Harrington and L. Jones. The irreducibility and monogenicity of power-compositional trinomials. arXiv preprint arXiv:2204.07784, 2022

    arXiv 2022

  12. [12]

    H. Hasse. Zahlentheorie.Akademie-Verlag, Berlin, 1963

    1963

  13. [13]

    A. Jakhar. On primes dividing the index of a quadrinomial.Rocky Mountain J. Math., 60(6):2117– 2125, 2020

    2020

  14. [14]

    Jakhar, R

    A. Jakhar, R. Kalwaniya, and P. Yadav. A study of monogenity of binomial composition.Acta Arithmetica, 221(4), 2025

    2025

  15. [15]

    Jakhar, S

    A. Jakhar, S. K. Khanduja, and N. Sangwan. On prime divisors of the index of an algebraic integer. Journal of Number Theory, 166:47–61, 2016

    2016

  16. [16]

    L. Jones. Monogenic polynomials with non-squarefree discriminant.Proceedings of the American Mathematical Society, 148(4):1527–1533, 2020

    2020

  17. [17]

    L. Jones. The monogenity of power-compositional eisenstein polynomials. InAnnales Mathematicae et Informaticae, volume 55, pages 93–113. Eszterh´ azy K´ aroly Egyetem L´ ıceum Kiad´ o, 2022

    2022

  18. [18]

    L. Jones. The monogenicity of power-compositional characteristic polynomials.Albanian Journal of Mathematics, 18(1):21–30, 2024

    2024

  19. [19]

    S. Kaur, S. Kumar, and L. Remete. On the index of power compositional polynomials.Finite Fields Appl., 107:102642 (20 pages), 2025

    2025

  20. [20]

    K. Kedlaya. A construction of polynomials with squarefree discriminants.Proceedings of the Amer- ican Mathematical Society, 140(9):3025–3033, 2012

    2012

  21. [21]

    S. K. Khanduja and B. Jhorar. When is r[θ] integrally closed?Journal of Algebra and Its Applications, 15(05):1650091, 2016

    2016

  22. [22]

    Sage Developers

    S. Sage Developers. The sage mathematics software system (version 9.0), 2020

    2020

  23. [23]

    Sharma and R

    H. Sharma and R. Sarma. Monogenity of composition of polynomials.The Ramanujan Journal, 67(2):45, 2025

    2025

  24. [24]

    K. Uchida. When isZ[α] the ring of the integers?Osaka Math. J., 14(1):155–157, 1977. (A. Choudhary)Department of Mathematics, Indian Institute of Science Education and Research, Pune, Maharashtra, India-411008 Email address:achoudhary1396@gmail.com (S. Pisolkar)Department of Mathematics, Indian Institute of Science Education and Research, Pune, Maharashtr...

    1977