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Statement

Setting (p. 1). An integer is $y$-smooth when each of its prime divisors is at most $y$. A polynomial $f\in\mathbb Z[t]$ of positive degree admits smoothness $\theta\ge0$ when $\lvert f(n)\rvert$ is $\lvert f(n)\rvert^\theta$-smooth for infinitely many integers $n$, and admits polysmoothness $\theta$ when some non-constant $g\in\mathbb Z[t]$ makes every irreducible factor of $f(g(t))$ of degree at most $\theta(\deg f)(\deg g)$. The paper remarks (p. 1) that polysmoothness $\theta$ implies smoothness $\eta$ for every $\eta>\theta$, by looking at the values $f(g(m))$ for large integers $m$.

Theorem 1.3 (p. 3, quoted). "Let $f\in\mathbb Z[t]$ be irreducible, and let $\alpha$ be a root of $f$ lying in its splitting field. Suppose that for some $\gamma\in\mathbb Q(\alpha)$ and $g\in\mathbb Z[t]$ of degree $k\geqslant2$, one has $\alpha=g(\gamma)$. Then $f(g(t))$ is divisible by the minimal polynomial of $\gamma$ over $\mathbb Q$, and hence $f$ admits polysmoothness $1-1/k$."

The paper reads Schinzel's construction (Acta Arith. 13 (1967), Lemma 10) as an instance of this theorem (pp. 3 and 7--8). After its direct proof, it notes that the theorem is also a special case of a proposition Schinzel attributes to Capelli (Proposition 3.1, p. 7).

Proof pointer

P. 7, where the proof takes $\deg f=d\geqslant2$. Since $f(g(\gamma))=f(\alpha)=0$, the minimal polynomial of $\gamma$ divides $f(g(t))$, and by Gauss's lemma an integral multiple of it of degree $d$ divides $f(g(t))$ in $\mathbb Z[t]$; the degree is $d$ because $\mathbb Q(\gamma)\subseteq\mathbb Q(\alpha)=\mathbb Q(g(\gamma))\subseteq\mathbb Q(\gamma)$. The cofactor has degree $kd-d$, so every factor has degree at most $(1-1/k)dk$.

Read depth

Claims checked: the statement was read clause by clause on the printed page and the short proof was followed step by step. Nothing here is independently reviewed.

Dependencies

None within the paper.

Source. J. W. Bober, D. Fretwell, G. Martin and T. D. Wooley, Smooth values of polynomials, J. Aust. Math. Soc. 108 (2020), no. 2, 245--261, doi:10.1017/S1446788718000320; the arXiv version 1 print (arXiv:1710.01970v1, 5 October 2017) is the edition read, and its labels and pages are cited here, as named on the source card.