Bounding univariate and multivariate reducible polynomials with restricted height

Abstract

Let $d,H \geqslant 2$, $m, u \geqslant 0$ be some integers satisfying $m+u \leqslant d$. Consider a set of univariate integer polynomials of degree d whose m coefficients for the highest powers of x and u coefficients for the lowest powers of x are fixed, whereas the remaining $g=d-m-u+1$ coefficients are all bounded by H in absolute value. We show that among those $(2H+1)^g$ polynomials at most $c d(2H+1)^{g-1}(\log (2H))^{\delta }$ are reducible over $\mathbb Q$, where the constant $c>0$ depends only on two extreme coefficients (if they are fixed) and does not depend on d and H. Here, $\delta =2$ if $m=u=0$; $\delta =1$ if only one of m, u is zero; $\delta =0$ if none of m, u is zero. This estimate is better than the previous one in certain range of d and H. We also prove an estimate for the number of integer reducible polynomials in $n \geqslant 2$ variables of degree $d \geqslant 1$ in each variable and height at most $H \geqslant 1$. It is completely explicit in terms of n, d, H and implies that the probability for such a polynomial to be reducible tends to zero as $\max (n,d,H) \rightarrow \infty$. The condition $n \geqslant 2$ is essential in the proof: despite some recent progress the problem in general remains open for $n=1$.