{"id":"openai-math/Squarefree-values-of-quartics-and-power-free-values-of-polynomials","pdf":"https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf","url":"https://github.com/openai/math/tree/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026","tags":["number-theory","squarefree-values","polynomials","density"],"score":0,"title":"Squarefree values of quartics and power-free values of polynomials","domain":"Number theory","family":"020","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":true,"family_title":"Squarefree quartics and power-free polynomial values"}