Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 020-01. Squarefree values of quartics and power-free values of polynomials
Manuscript map · lines 2895–2904
OpenAI@2026-10-07#rem:oai-p-020-01
Authors: OpenAI. Family 020.
Title: Squarefree values of quartics and power-free values of polynomials.
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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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