On truncations of valuations

In this paper we study the truncation $\nu_q$ of a valuation $\nu$ on a polynomial $q$. It is known that when $q$ is a key polynomial, then $\nu_q$ is a valuation. It is also known that the converse does not hold. We show that when $q$ is a key polynomial, then $\nu_q$ is the restriction of the truncation given by an optimizing root of $q$. We also discuss which conditions assure that $\nu_q=\nu$. Finally, we assume that $\nu_q$ is a valuation and present some conditions, given in terms of the graded algebra, to assure that $q$ is a key polynomial.