Bober 2020 smooth values polynomials

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corollary_1_2: Bober, Fretwell, Martin and Wooley's corollary: for every epsilon > 0 and every quadratic f in Z[t] there are infinitely many natural numbers n with every prime factor of f(n) at most n^epsilon.

corollary_1_4: Bober, Fretwell, Martin and Wooley's corollary of their field-theoretic criterion: an irreducible f in Z[t] of the form g(h(t)) - t, with g and h integer polynomials of degree exceeding 1, has f(g(t)) divisible by the minimal polynomial of h(alpha) and admits polysmoothness 1 - 1/deg(g).

theorem_1_1: Bober, Fretwell, Martin and Wooley's main theorem: for a quadratic f in Z[t] there are integer polynomials g of arbitrarily large odd degree k with f(g(t)) a product of polynomials of degree at most c k over the square root of log log k.

theorem_1_3: Bober, Fretwell, Martin and Wooley's field-theoretic criterion: if a root alpha of an irreducible f in Z[t] equals g(gamma) for some gamma in Q(alpha) and some g in Z[t] of degree k at least 2, then the minimal polynomial of gamma divides f(g(t)) and f admits polysmoothness 1 - 1/k.

theorem_1_5: Bober, Fretwell, Martin and Wooley's theorem on trinomials: for an integer k at least 2 and integers a, b, the polynomial t^k + a t^(k-1) - b with b nonzero admits polysmoothness phi(k-1)/(k-1), and a t^k - t + b with ab nonzero admits polysmoothness phi(k)/k.

theorem_2_1: Bober, Fretwell, Martin and Wooley's cyclotomic construction: for a product f of l binomials a_j t^(k_j) - b_j with a_1 ... a_l nonzero there are integer polynomials g of arbitrarily large degree d with f(g(t)) a product of polynomials of degree at most c d/(log log d)^(1/l).

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Bober, J. W. and Fretwell, D. and Martin, G. and Wooley, T. D., Smooth values of polynomials. J. Aust. Math. Soc. 108 (2020), no. 2, 245--261. doi:10.1017/S1446788718000320. The copy read for this card is arXiv version 1 (5 October 2017). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1710.01970), every other right reserved.

The paper asks whether every f in Z[t] of positive degree admits polysmoothness epsilon for every epsilon > 0, where f admits polysmoothness theta when some non-constant g in Z[t] makes every irreducible factor of f(g(t)) of degree at most theta deg(f) deg(g), and answers it affirmatively for degree two. Theorem 1.1 shows that for quadratic f there are c > 0 and g of arbitrarily large odd degree k such that f(g(t)) factors into polynomials each of degree at most c k/(log log k)^{1/2}, so f admits polysmoothness epsilon for every epsilon > 0; Corollary 1.2 deduces that for each epsilon > 0 there are infinitely many n with f(n) being n^epsilon-smooth. The paper calls Schinzel's half-century-old exponent 0.2795... the sharpest earlier result for quadratics, says Theorem 1.1 supersedes it for degree two, and names quadratics such as 4t^2+4t+9 as out of reach of Schinzel's methods. Theorem 2.1 does the same for every product f(t) = prod_{j=1}^l (a_j t^{k_j} - b_j) of binomials with integers a_j, b_j, k_j, k_j >= 1 and a_1...a_l != 0: for some c = c(k_1,...,k_l) > 0 there are g of arbitrarily large degree d with factor degrees at most c d/(log log d)^{1/l}. Theorem 1.3 gives a general algebraic mechanism: if f is irreducible with root alpha and alpha = g(gamma) for gamma in Q(alpha) and g in Z[t] of degree k >= 2, then the minimal polynomial of gamma divides f(g(t)) and f admits polysmoothness 1-1/k, with Corollary 1.4 the case f(t) = g(h(t)) - t. Theorem 1.5 treats trinomials: for a natural number k >= 2 and integers a, b, t^k + a t^{k-1} - b with b != 0 admits polysmoothness phi(k-1)/(k-1), and a t^k - t + b with ab != 0 admits phi(k)/k; for t^k - t - 1 the bound phi(k)/k tends to 0 along k equal to the product of the first n primes, and the proof uses cyclotomic factorization. Theorem 3.2 answers a question of Granville and Pleasants for irreducible cubics, and Theorem 6.1 gives a criterion obstructing reducible compositions f(g(t)) with g quadratic, illustrated by sextics. Erdos problem 369 asks for k consecutive n^epsilon-smooth integers up to n. The paper does not discuss runs of consecutive integers; Theorem 2.1 covers the product (t+1)(t+2)...(t+k) of k consecutive linear polynomials, and the problem's site records a deduction from Theorem 2.1, pointed out by Wooley, of a run of k consecutive n^epsilon-smooth integers in [n - n^c, n] for some c < 1 and all large n.

Source: https://arxiv.org/abs/1710.01970.

Bears on. #369: Theorem 2.1 (p. 5) applies to $f(t)=(t+1)(t+2)\cdots(t+k)$, the shape (2.1) with $l=k$, every $a_j=k_j=1$ and $b_j=-j$, giving integer polynomials $g$ of arbitrarily large degree $d$ with $f(g(t))$ a product of polynomials of degree at most $cd/(\log\log d)^{1/k}$. The paper does not discuss runs of consecutive integers or the problem.

Results. Pages and labels are those of the arXiv version 1 print read.

• Theorem 1.1 (p. 1): for quadratic $f\in\mathbb Z[t]$ there are $c>0$ and $g\in\mathbb Z[t]$ of arbitrarily large odd degree $k$ with $f(g(t))$ a product of polynomials of degree at most $ck/\sqrt{\log\log k}$; so $f$ admits polysmoothness $\varepsilon$ for every $\varepsilon>0$.
• Corollary 1.2 (p. 2): for $\varepsilon>0$ and quadratic $f\in\mathbb Z[t]$ there are infinitely many $n\in\mathbb N$ with $f(n)$ $n^\varepsilon$-smooth.
• Theorem 1.3 (p. 3): if $f\in\mathbb Z[t]$ is irreducible with root $\alpha$, and $\alpha=g(\gamma)$ with $\gamma\in\mathbb Q(\alpha)$ and $g\in\mathbb Z[t]$ of degree $k\geqslant2$, then the minimal polynomial of $\gamma$ divides $f(g(t))$ and $f$ admits polysmoothness $1-1/k$.
• Corollary 1.4 (p. 3): if $f\in\mathbb Z[t]$ is irreducible with root $\alpha$ and $f(t)=g(h(t))-t$ with $g,h\in\mathbb Z[t]$ of degree exceeding $1$, then the minimal polynomial of $h(\alpha)$ divides $f(g(t))$ and $f$ admits polysmoothness $1-1/\deg(g)$.
• Theorem 1.5 (p. 4): for a natural number $k\geqslant2$ and $a,b\in\mathbb Z$, $t^k+at^{k-1}-b$ with $b\ne0$ admits polysmoothness $\phi(k-1)/(k-1)$, and $at^k-t+b$ with $ab\ne0$ admits polysmoothness $\phi(k)/k$.
• Theorem 2.1 (p. 5): for $f(t)=\prod_{j=1}^l(a_jt^{k_j}-b_j)$ with integers $a_j,b_j,k_j$, $k_j\geqslant1$ and $a_1\cdots a_l\ne0$, there are $c=c(k_1,\ldots,k_l)>0$ and $g\in\mathbb Z[t]$ of arbitrarily large degree $d$ with $f(g(t))$ a product of polynomials of degree at most $cd/(\log\log d)^{1/l}$; so $f$ admits polysmoothness $\varepsilon$ for every $\varepsilon>0$.

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