Squarefree quartics and power-free polynomial values
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
Squarefree values of quartics and power-free values of polynomials
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≤ d\le8. The proof combines number-field factorization, determinant estimates with…
Full source 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≤ 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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