Family 020Number theoryLeanNot yet digested
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 -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.
Manuscripts in this family · 1
Squarefree values of quartics and power-free values of polynomials
Sep 24, 2026·manuscript.pdfLean
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 -power-free density in degrees . 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 -power-free density for every d ≥ 4.