# Key Polynomials in dimension 2

## TL;DR

This paper establishes a bijection between valuations of a two-dimensional regular local ring and those of its quotient field, and provides a new proof of the valuation set forming a non-metric tree structure.

## Contribution

It introduces a bijection between valuations of $Quot(R)$ and $k(x,y)$, and offers a new proof of the non-metric tree structure of valuations centered at $R$.

## Key findings

- Bijection between valuations of $Quot(R)$ and $k(x,y)$.
- Valuations form a non-metric tree structure.
- New proof of valuation set structure.

## Abstract

Let $R$ be a two-dimensional regular local ring. In this paper, we prove that there is a bijection between the set of all valuations of $Quot(R)$ centered at $R$ and valuations of $k(x,y)$ centered at $k[x,y]_{(x,y)}$, where $k$ is the residue field of $R$ and $x$ and $y$ are independent variables. Moreover, we give a new proof, for the fact that the set of all normalized real valuations centered at $R$ admits a structure of non metric tree.