On the distribution of reducible polynomials

TL;DR

This paper investigates the asymptotic behavior of reducible polynomials over integers, establishing precise growth rates for their counts based on degree and height, especially for quadratic and higher degrees.

Contribution

It determines the exact asymptotic order of the number of reducible polynomials over integers for degrees two and higher, providing new insights into their distribution.

Findings

For degree 2, the count grows as t^2 log t.

For degrees greater than 2, the count grows as t^n.

Identifies the size of specific subsets of reducible polynomials.

Abstract

Let Y_n(t) denote the set of all polynomials over the ring Z which are reducible over the field Q and of degree n>1 and of height not greater than t. We show that the true order of magnitude of |Y_n(t)| equals t^2 log t in the special case n=2 and it equals t^n for each n>2. We also determine the true order of magnitude of the size of certain interesting subsets of Y_n(t).