Quartic Thue equations

Abstract

We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper upper bounds.

Upper Bounds for the Number of Solutions to Quartic Thue Equations

“…We will slightly modify those ideas and use them to give an upper bound for the solutions of (1) with |y| < M (F ) 3.5 . Then we will improve the main result in [2] by giving better upper bounds for the number of solutions (x, y) with large |y| ≥ M (F ) 3.5 to equation (1). The following is the main result of this manuscript.…”

“…Let M (F ) be the Mahler measure of F (x, y). In [2] we used some ideas of Stewart [16] to bound the number of solutions with |y| < M (F ) 6 . We will slightly modify those ideas and use them to give an upper bound for the solutions of (1) with |y| < M (F ) 3.5 .…”

“…Suppose that F (x, y) is reducible and can be factored over Z[x, y] as follows F (x, y) = F 1 (x, y)F 2 (x, y), with deg(F 1 ) ≤ deg(F 2 ) and F 1 irreducible over Z [x, y]. Therefore, the following equations must be satisfied: (2) F 1 (x, y) = ±1 and (3)…”

“…In this manuscript, we give new and sharper bounds for the number of solutions to equation (1). The bound given in [2] is improved here mostly due to some adjustment in the definition of the logarithmic curve φ(x, y) in Section 6. We also study the geometry of binary forms with respect to their signature to get amore precise understanding of the distribution of solutions to (1).…”

“…We also appeal to a result of Voutier (Proposition 2.2) to estimate the height of algebraic numbers in the number field generated over Q by a root of the equation F (x, 1) = 0. These allow us to extend our method introduced in [2] for quartic forms that split in R to all quartic forms.…”