Heights of algebraic points lying on curves or hypersurfaces

Our first aim will be to give an explicit version of a generalization of the results of Zhang and Zagier on algebraic points $(x,y)$ with $x+y+ 1 = 0$. Secondly, we will show that distinct algebraic points lying on a given curve of certain type can be distinguished in terms of some height functions. Thirdly, we will derive a bound for the number of points on such a curve whose heights are under a given bound and whose coordinates lie in a multiplicative group of given rank.