Counting squarefree discriminants of trinomials under abc

TL;DR

Under the assumption of the abc conjecture, the paper proves that a positive proportion of trinomials of the form t^n + a t + b have irreducible and squarefree discriminants for odd n ≥ 5.

Contribution

It establishes, assuming abc, that many such trinomials have irreducible and squarefree discriminants, linking polynomial irreducibility and discriminant properties.

Findings

Positive proportion of trinomials with squarefree discriminants

Irreducibility of trinomials under abc conjecture

Results valid for odd n ≥ 5

Abstract

For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their discriminants are squarefree.