On truncations of valuations

Abstract

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