In this paper the authors obtain some estimations for intersection points of two maps \(f,g:X\to Y\). Instead of considering intersection points on the target space \(Y\) they focus on common value pairs, i.e the pairs \((u,v)\) in the domain \(X^2\) of \(f\times g\) with \(f(u)=g(v)\). They apply the idea of Nielsen root theory to such a general case: preimage of a set.