Valuation overrings of polynomial rings and group of divisibility

In this work, we discuss the types of valuation overrings of $K[x_{1}, x_{2}, ..., x_{n}]$ based on the rank and rational rank of value groups. Also, we describe the group of divisibility of a finite intersection of valuation overrings of $K[x_{1}, x_{2}, ..., x_{n}].$ In particular, we focus on the case for $n > 3.$