Bounding univariate and multivariate reducible polynomials with restricted height (Q2416505)

For integers \(d\ge 2\) and \(H\ge 2\), let \(\mathbb{P}(d,H)\) (and \(\mathbb{P}^*(d,H))\) denote the probability that a univariate (and monic) polynomial of degree \(d\), with integer coefficients of absolute value at most \(H\), is irreducible over \(\mathbb{Q}\). The author gives simple proofs for the completely explicit estimates \[ \mathbb{P}(d,H)< \frac{1+d\bigl(\log (2H)\bigr)^2}{2H+1},\quad\mathbb{P}^*(d,H)< \frac{1+d \log (2H)}{2H+1}. \] Similar bounds are found when several first and several last coefficients of the polynomial are fixed. A slightly more complicated estimate is valid for the probability that a multivariate integer polynomial of degree $d$ in each variable and height \(H\) is irreducible. In its proof, a key tool is a lemma of independent interest, which asserts that certain specialization at some \(n.1\) variables of a random polynomial in \(n\ge 2\) variables of degree \(d\) in each variable is a univariate polynomial of degree \(d\) with uniformly distributed coefficients.