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.
