In the 1980's we developed an effective specialization method and used it to prove effective finiteness theorems for Thue equations, decomposable form equations and discriminant equations over a restricted class of finitely generated domains (FGD's) over $\mathbb{Z}$ which may contain not only algebraic but also transcendental elements. In 2013 we refined with Evertse the method and combined it with an effective result of Aschenbrenner (2004) concerning ideal membership in polynomial rings over $\mathbb{Z}$ to establish effective results over arbitrary FGD's over $\mathbb{Z}$. By means of our method general effective finiteness theorems have been obtained in quantitative form for several classical Diophantine equations over arbitrary FGD's, including unit equations, discriminant equations (Evertse and Gyory, 2013, 2017), Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation (Bérczes, Evertse and Gyory, 2014), generalized unit equations (Bérczes, 2015), and the Catalan equation (Koymans, 2015). In the first part of the talk we shall briefly survey these results. Recently we proved with Evertse effective finiteness theorems in quantitative form for norm form equations, discriminant form equations and more generally for decomposable form equations over arbitrary FGD's. In the second part, these new results will be presented. Some applications will also be discussed.[-]

In a recent paper, Istvan Gaal and Laszlo Remete studied the integer solutions to binary quartic Thue equations of the form $x^4-dy^4 = \pm 1$, and used their results to determine pure quartic number fields which contain a power integral basis. In our talk, we propose a new way to approach this diophantine problem, and we also show how an effective version of the abc conjecture would allow for even further improvements. This is joint work with M.A. Bennett. We also discuss a relation between this quartic diophantine equation to recent joint work with P.-Z. Yuan.[-]