On the reducibility type of trinomials

Say a trinomial x^n+A x^m+B \in \Q[x] has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in \Q[x] of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq ... \leq n_k$. Specifying the reducibility type of a monic polynomial of fixed degree is equivalent to specifying rational points on an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable hope that all its rational points may be described; and techniques are available that may also find all points when the genus is 2. Thus all corresponding reducibility types may be described. These low genus instances are the ones studied in this paper.Comment: to appear in Acta Arithmetic

Smooth values of polynomials

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in\mathbb{Z}[t]$ and any $\epsilon > 0$, there are infinitely many $n\in\mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\epsilon}$

Probabilistic Galois Theory

We show that there are at most $O_{n,\epsilon}(H^{n-2+\sqrt{2}+\epsilon})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_{n}(H^{n-1/2}\log H)$.Comment: 10 page

On the smallest number of terms of vanishing sums of units in number fields

L