Squarefree values of irreducible quartic polynomials, and (d-2)-power-free values in every degree d at least 4

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Comparator statement: PowerFreeValues.lean evidence for this event

Squarefree values of irreducible quartic polynomials, and (d-2)-power-free values in every degree d at least 4 evidence for this event

Theorem 1.1: for $f\in\mathbb Z[x]$ irreducible over $\mathbb Q$ of degree $4\le d\le8$ with no fixed prime $(d-2)$th-power divisor, the number of $n\le X$ with $f(n)$ $(d-2)$-free is $c_{f,d-2}X+o_f(X)$ with the Euler-product constant, which is positive; for $d=4$ this is squarefree values of quartics, e.g. $n^4+2$ and $n^4+1$. With Browning's theorem for $d\ge9$ (Corollary 1.2) the $(d-2)$-free density holds for all $d\ge4$; Corollary 1.3 treats separable products. Not shown: a power-saving error term, uniformity in $f$, or squarefree values in degree five and above. Priority: Carella's preprint claims $n^4+1$ and $n^4+2$, and Zapata Ceballos-Jalalvand claim positive density for a class including $n^4+1$; the manuscript uses neither. parent of this event

VibeMathed record: Squarefree values of irreducible quartic polynomials, and (d-2)-power-free values in every degree d at least 4 evidence for this event

Squarefree values of irreducible quartic polynomials, and (d-2)-power-free values in every degree d at least 4 parent of this event

Erdos, Arithmetical properties of polynomials (1953) evidence for this event