MacLane–Vaquié chains of valuations on a polynomial ring

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions $v$ to valuations the polynomial ring $K[x]$. introduce certain MacLane-Vaqui\'e chains residually transcendental valuations, we prove that every valuation $\mu$ $K[x]$ is limit finite or countably infinite chain. This chain underlying essentially unique contains arithmetic data yielding an explicit description graded algebra as over $v$.