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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 1.1 (p. 1, quoted). "Let $f\in\mathbb Z[t]$ be quadratic. Then for some $c>0$ there are polynomials $g\in\mathbb Z[t]$ of arbitrarily large odd degree $k$ for which $f(g(t))$ factors as a product of polynomials of degree at most $ck/\sqrt{\log\log k}$. Thus $f$ admits polysmoothness $\varepsilon$ for any $\varepsilon>0$."

The paper answers, for degree two, the question it poses on p. 1: whether every $f\in\mathbb Z[t]$ of positive degree admits polysmoothness $\varepsilon$ for every $\varepsilon>0$. It notes (p. 3) that the theorem supersedes, for quadratics, Schinzel's polysmoothness exponent $\theta(2)=0.27950849\ldots$.

Proof pointer

A quadratic that is a product of two linear factors is the case $l=2$, $k_1=k_2=1$ of Theorem 2.1 (end of Section 2, p. 6). For irreducible $f=at^2+bt+c$ (Section 4, pp. 10--12), $k$ is taken to be the product of the primes below a large $X$ not dividing $2a\phi(a)$, so that $\prod_{p\mid k}(1-1/p)$ is of order $1/\log\log k$. Lemma 4.1 (p. 10) supplies integers with $(ma\alpha+n)^k=A\alpha+B$, $A\ne0$, $(A,B)=1$, for a root $\alpha$ of $f$. Then $f((t^k-B)/A)$ splits over $\mathbb Q$ into a constant times polynomials $h_d$ of degree $2\phi(d)$, one for each $d\mid k$, built from $d$-th roots of unity. Since $B$ is a $k$-th power $z^k$ modulo $A$, the shift $g(t)=((At+z)^k-B)/A$ has integer coefficients and odd degree $k$, and the factors of $f(g(t))$ have degree at most $2k\prod_{p\mid k}(1-1/p)$.

Read depth

Claims checked: the definitions and the statement were read clause by clause on the printed pages; the proof was read for its structure, not checked line by line. Nothing here is independently reviewed.

Dependencies

Lemma 4.1 (p. 10) and Theorem 2.1 (p. 5) of the same 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.