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. (Lean)

Manuscript

Squarefree values of quartics and power-free values of polynomials

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026}

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.

Result 020 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.