Northcott numbers for generalized weighted Weil heights

We give a generalization of weighted Weil heights. These heights generalize both Weil’s heights and Dobrowolski’s height. We study Northcott numbers for our heights. Our results generalize the authors’ former work on Vidaux and Videla’s question about the Northcott number. As an application, we evaluate Northcott numbers for Talamanca’s spectral height on matrices.