Quartic polynomials with a given discriminant

Abstract

Let $ 0\ne D \in \Bbb Z$ and let $Q_D$ be the set of all monic quartic polynomials $ x^4 +ax^3 +bx^2 + cx + d \in \Bbb Z[x]$ with the discriminant equal to $D$. In this paper we will devise a method for determining the set $Q_D$. Our method is strongly related to the theory of integral points on elliptic curves. The well-known Mordell's equation plays an important role as well in our considerations. Finally, some new conjectures will be included inspired by extensive calculation on a computer.Let $ 0\ne D \in \Bbb Z$ and let $Q_D$ be the set of all monic quartic polynomials $ x^4 +ax^3 +bx^2 + cx + d \in \Bbb Z[x]$ with the discriminant equal to $D$. In this paper we will devise a method for determining the set $Q_D$. Our method is strongly related to the theory of integral points on elliptic curves. The well-known Mordell's equation plays an important role as well in our considerations. Finally, some new conjectures will be included inspired by extensive calculation on a computer.