  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.
