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Squarefree values of quartics and power-free values of polynomials
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Abstract
We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding $(d-2)$-power-free density in degrees $4\le d\le8$. The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the $(d-2)$-power-free density for every d ≥ 4.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- erdosproblems.com forum (T. Bloom, M. Firsching and others), Erdős Problem #978 discussion thread (forum, Sep 2025 – Mar 2026). This thread predates the paper and does not comment on it, but discusses the hypotheses of the question that Theorem 1.1 addresses: Firsching reports a degree-6 counterexample with fixed divisor $2^4$, and the problem statement was then amended to exclude fixed prime-power divisors.
Formalization
- OpenAI / mathcommunity, Lean scope notes for family 020: Squarefree quartics and power-free polynomial values (lean/docs/020.md). The release's scope note: the $(d-2)$-free Euler-product density for every irreducible $f\in\mathbb Z[x]$ of degree $d\ge4$ with no fixed prime $(d-2)$th-power divisor, with no monicity, primitivity or height restriction. [family-level]
- OpenAI, lean/OAI/NumberTheory/PowerFree/Main.lean (openai/math, commit adc7f12). Contains the theorems
density($4\le d\le8$),allDegrees($d\ge4$) and the quartic dyadic tail boundquartic_dyadic_tail(saving $N^{-3/1000}$). [family-level] - OpenAI, lean/ComparatorChallenges/PowerFreeValues.lean (openai/math, commit adc7f12). The comparator statement of
allDegrees, definingPowerFree,rho,LocallyAdmissible,densityConstantandDensityStatement. [family-level] - Google DeepMind formal-conjectures, FormalConjectures/ErdosProblems/978.lean (formal-conjectures). Formalizes the statement of Erdős Problem #978, whose parts (ii) and (iii) cite this paper as [OAI26a] and link the openai/math Lean proof; the file itself contains only
sorryplaceholders. [family-level]
Related prior literature mentioned by others
- S. R. Zapata Ceballos, F. Jalalvand, On the Squarefree Values of Degree-2q Polynomials (arXiv:2608.10335, v1 11 Aug 2026; v2 15 Oct 2026). The paper's introduction cites v1 as a claim that includes $n^4+1$; the authors withdrew it in v2 with the comment "The proof of Proposition 2.3 is incomplete". [cites earlier version]
- Theodor Estermann, Einige Sätze über quadratfreie Zahlen (reference in Bogdan Grechuk's exposition) (Mathematische Annalen 105, 653–662, 1931). Grechuk cites this work on the quadratic case as historical context for the paper's quartic squarefree-value theorem.
Possibly relevant literature (found by the agent, not checked)
- B. Poonen, Squarefree values of multivariable polynomials (Duke Math. J. 118 (2003), 353–373). Extends the $abc$-conditional squarefree density, which the introduction attributes to Granville for univariate polynomials, to multivariable polynomials.
- T. D. Browning, A. Gorodnik, Power-free values of polynomials on symmetric varieties (arXiv:1606.06342, 2016/2017). An asymptotic count of $r$-free polynomial values at integral points of symmetric varieties, a broader setting for the paper's power-free density problem.
Digests and expositions
- Bogdan Grechuk, Square-free values of quartic polynomials (Functor Network, 2026-10-14). Explains the local density factors, the quartic examples, the large-prime obstacle and determinant-method strategy, and the higher-degree consequence via Browning's theorem.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 020
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026,
author = {{OpenAI}},
title = {{Squarefree values of quartics and power-free values of polynomials}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf}{OAI:Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/020.md. The Lean statements have not been checked by people against the paper's claims.
Family 020: Squarefree quartics and power-free polynomial values
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d-2)$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.
This is the only paper in this family.