Quartic Polynomials with a Given Discriminant

Abstract

AbstractLet 0 ≠D∈ ℤ and letQDbe the set of all monic quartic polynomialsx4+ax3+bx2+cx+d∈ ℤ[x] with the discriminant equal toD. In this paper we will devise a method for determining the setQD. 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 calculations on a computer.