We shall give a new treatment to intersection points of two maps, named common value pairs. Given two maps $f,g\colon X\to Y$. Instead of considering intersection points on target space $Y$, we focus on the pairs in the domains $X$, the pair $(u,v)$ with $f(u)=g(v)$. The set of all these pairs is exactly the preimage of product $f\times g$ at the diagonal in $Y^2$. We shall apply the idea of Nielsen root theory into such a general case: preimage of a set. Hence, some estimation for common value pairs and therefore for intersection points are obtained.
"Common value pairs and their estimations." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 725 - 739, december 2017. https://doi.org/10.36045/bbms/1515035018