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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 $m$.

Shape (2.1) (p. 5). For integers $a_j,b_j,k_j$ with $k_j\geqslant1$ ($1\leqslant j\leqslant l$),

$$f(t)=\prod_{j=1}^{l}\bigl(a_jt^{k_j}-b_j\bigr).$$

The print does not define $\mathbf k$; it is read here as $(k_1,\ldots,k_l)$.

Theorem 2.1 (p. 5, quoted). "Let $f\in\mathbb Z[t]$ be a polynomial of the shape (2.1) with $a_1\cdots a_l\neq0$. Then for some $c=c(\mathbf k)>0$, there are polynomials $g\in\mathbb Z[t]$ of arbitrarily large degree $d$ for which $f(g(t))$ factors as a product of polynomials of degree at most $cd/(\log\log d)^{1/l}$. Thus $f$ admits polysmoothness $\varepsilon$ for any $\varepsilon>0$."

The paper says the argument is a modification of Balog and Wooley's proof of Lemma 2.2 of their 1998 paper on strings of consecutive integers with no large prime factors (p. 5).

Proof pointer

Pp. 5--6. With $k=k_1\cdots k_l$ and $y$ large, Lemma 2.1 of Balog and Wooley (1998) splits the primes up to $y$ coprime to $k$ into $l$ sets $\mathcal P_i$, each with $\prod_{p\in\mathcal P_i}(1-1/p)$ less than $2\bigl(k/(\phi(k)\log y)\bigr)^{1/l}$ and product of its primes less than $y^2e^{5y/(4l)}$. With $\gamma_i$ the product of the primes of $\mathcal P_i$ and $\Gamma=\gamma_1\cdots\gamma_l$, exponents chosen by congruences modulo the $\gamma_j$ give a monomial $g(t)=t^\Gamma\prod_ja_j^{\lambda_j}b_j^{\mu_j}$ of degree $\Gamma$ for which each $a_jg(t)^{k_j}-b_j$ equals $b_j(z_j^{\gamma_j}-1)$ for a monomial $z_j$ of degree $k_j\Gamma/\gamma_j$. Splitting $z_j^{\gamma_j}-1$ into cyclotomic polynomials gives factors of degree at most $\Gamma\max_jk_j\phi(\gamma_j)/\gamma_j$, which is $\ll\Gamma(\log\log\Gamma)^{-1/l}$ since $y\asymp\log\Gamma$.

Reading note: the exponents $\mu_j$ of the printed construction are positive, so as printed $g$ vanishes identically when some $b_j=0$; the statement itself does not exclude $b_j=0$.

Read depth

Claims checked: the statement was read clause by clause on the printed page; the proof was read for its structure, and the cited lemma of Balog and Wooley was not checked here. Nothing here is independently reviewed.

Dependencies

Lemma 2.1 of A. Balog and T. D. Wooley, On strings of consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 64 (1998), no. 2, 266--276, whose card is balog_1998_strings_consecutive_integers_no_large_prime_factors.

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.

Bears on

• Problem 369: the product $f(t)=(t+1)(t+2)\cdots(t+k)$ of $k$ consecutive linear polynomials has the shape (2.1) with $l=k$, every $a_j=k_j=1$ and $b_j=-j$, so the theorem applies to it. The paper does not discuss runs of consecutive integers or the problem.