Of limit key polynomials (Q2039729)

Let \(K\) be a field and \(v\) a valuation on the polynomial ring \(K[x]\), with value group \(\Gamma_v\). For each \(\gamma\in \Gamma_v\), we have the following abelian groups \(\mathcal{P}_{\gamma}^+=\{ g\in K[x]; \mu(g)>\gamma\}\subset\mathcal{P}_{\gamma}=\{ g\in K[x]; \mu(g)\geq\gamma\}\). The graded algebra \(gr_v(K[x])=\oplus_{\gamma\in\Gamma_v}\mathcal{P}_{\gamma}/ \mathcal{P}_{\gamma}^+\) is an integral domain. A MacLane-Vaquie (MLV) key polynomial for \(v\) is a monic polynomial \(\phi\in K[X]\) whose initial term generates a prime ideal in \(gr_v(K[x])\), which cannot be generated by the initial term of a polynomial of smaller degree. The abstract key polynomials for \(v\) are defined in a technical way. In the paper under review, the authors try to find relations between the MLV key polynomials for valuations \(\mu\leq v\) and the abstract key polynomials for \(v\).