On truncations of valuations (Q2069815)

Let \(K\) be a field, \(\overline{K}\) be an algebraic closure of \(K\), \(\nu\) be a pseudo-valuation on the polynomial ring \(K[x]\) and \(\mu\) be a pseudo-valuation which extends \(\nu\) to \(\overline{K}[x]\). The purpose of this paper is to prove three results. Before to state them, we have to define the valuation-transcendental pseudo-valuations, the mappings \(\nu_q\) and the key polynomials. The pseudo-valuation \(\nu\) is called valuation-transcendental if either it is not a Krull valuation, or \(\nu (K(x))/\nu K\) is not a torsion group, or the residue field extension \(K(x)\nu | K\nu\) is not an algebraic extension. For any \(q\in K[x]\), every \(f\in K[x]\) can be written in a unique way as \(f=f_0+f_1 q+\cdots +f_n q^n\), where for \(0\leq i\leq n\) we have either \(f_i=0\) or \(\mbox{deg}(f_i)<\mbox{deg}(q)\). Then we denote by \(\nu_q(f)\) the minimum of \(\nu(f_0),\nu(f_1q),\dots,\nu(f_nq^n)\), and by \(\delta_q(f)\) the greatest \(i\in \{0,\dots,n\}\) such that \(\nu_q(f)=\nu(f_iq^i)\). For \(f \in K[x]\) and \(r\in \mathbb{N}\backslash \{0\}\) we denote by \(\partial_r(f)\) the \(r\)-th formal derivative of \(f\). Let \(\epsilon(f)\) be the maximum of the \(\displaystyle{\frac{\nu(f)-\nu(\partial_r f)}{r}}\)'s, where \(r\) runs over \(\mathbb{N}\backslash \{0\}\). A key polynomial is a monic polynomial \(Q \in K[x]\) such that \(\forall f \in K[x] \; (\epsilon(f) \geq \epsilon(Q)\Rightarrow \mbox{deg}(f) \geq\mbox{deg}(Q))\). If \(Q\) is a key polynomial, then \(\nu_Q\) is a pseudo-valuation. The second result states that \(\nu\) is valuation-transcendental if, and only if, there exists a key polynomial \(Q\in K[x]\) such that \(\nu=\nu_Q\). For every \(f\in K[x]\) we let \(\delta(f)\) be the maximum of the \(\mu(x-a)\)'s, where \(a\) runs over the set of roots of \(f\) in \(\overline{K}\). An optimizing root for \(f\) is a root \(a\in \overline{K}\) such that \(\mu(x-a)=\delta(f)\). The first result states that if \(\nu\) is a Krull valuation, \(Q\) is a key polynomial and \(a\) is an optimizing root for \(Q\), then the restriction of \(\mu_{x-a}\) to \(K[x]\) is equal to \(\nu_Q\). Finally, let \(q\in K[x]\) be a polynomial such that \(\nu_q\) is a pseudo-valuation. Denote by \(\mathcal{G}_q\) the graded algebra associated to \(\nu_q\): \(\mathcal{G}_q=\bigoplus_{\gamma\in \nu_q(K[x])} \{f\in K[x] \backslash \{0\} \; :\; \nu_q(f)\geq \gamma\}/\{f\in K[x] \backslash \{0\} \; :\; \nu(f)> \gamma\}.\) For \(f\in K[x]\) such that \(\nu_q(f)<+\infty\), we denote by \(\mathrm{in}_q(f)\) its image in \(\{g\in K[x] \backslash \{0\} \; :\; \nu_q(g)\geq \nu_q(f)\}/\{g\in K[x] \backslash \{0\} \; :\; \nu_q(g)> \nu_q(f)\},\) and we let \(R_q\) be the additive subgroup of \(\mathcal{G}_q\) generated by the \(\mathrm{in}_q(f)\)'s, where \(f\in K[x]\) and \(\mbox{deg}(f)<\mbox{deg}(q)\). The third result states that \(q\) being a key polynomial is equivalent to both of the following three properties : (i) \(R_q\) is a subring of \(\mathcal{G}_q\), (ii) for every \(f\), \(g\) in \(K[x]\), \(\delta_q(fg)=\delta_q(f)+\delta_q(g)\), (iii) for every \(f\), \(g\) in \(K[x]\), \(\delta_q(f)=\delta_q(g)=0\Rightarrow \delta_q(fg)=0\).