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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 1.5 (p. 4). Let $k\geqslant2$ be a natural number.

• (i) If $f(t)=t^k+at^{k-1}-b$ with $a,b\in\mathbb Z$ and $b\ne0$, then $f$ admits polysmoothness $\phi(k-1)/(k-1)$.
• (ii) If $f(t)=at^k-t+b$ with $a,b\in\mathbb Z$ and $ab\ne0$, then $f$ admits polysmoothness $\phi(k)/k$.

No irreducibility is assumed. The paper illustrates (ii) with $f_k(t)=t^k-t-1$, irreducible for every $k\geqslant2$ by Selmer: taking $k$ to be the product of the first $n$ primes and letting $n\to\infty$, the exponent $\phi(k)/k$ tends to $0$ (p. 4).

Proof pointer

Pp. 12--13, by cyclotomic factorization. For (i), with $g(t)=b^kt^{k-1}-a$ one has $f(g(t))=b\bigl((btg(t))^{k-1}-1\bigr)$, a constant times $\prod_{d\mid k-1}\Phi_d(btg(t))$, whose factors have degree at most $\max_{d\mid k-1}\phi(d)k$ out of $k(k-1)$. For (ii), with $g(t)=a^{k+1}t^k+b$ one has $f(g(t))=a\bigl(g(t)^k-(at)^k\bigr)$, which splits as $a\prod_{d\mid k}(at)^{\phi(d)}\Phi_d(g(t)/(at))$ into factors of degree $k\phi(d)$ out of $k^2$.

Read depth

Claims checked: the statement was read clause by clause on the printed page and the two factorizations were followed. 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.