A note on points on algebraic sets

In this short note, we count the points on algebraic sets which lie in a subset of a domain. It is proved that the set of points on algebraic sets coming from certain subsets of a domain has the full asymptotic. This generalizes the first theorem of [E. Alkan and E.S. Yörük, Statistics and characterization of matrices by determinant and trace, Ramanujan J., 2019] and also anwers some questions from the same article.