Discrete valuations on Weyl skew fields (Q1355635)

This paper begins a study of the \(k\)-valuations of \(D_1(k)\), the skew-field of fractions of the first Weyl algebra \(A_1(X,Y)\). Attention is restricted to discrete \(k\)-valuations which are compatible with the Bernstein filtration of \(D_1(k)\) defined by the total degree in \(X\) and \(Y\) and such that the graded algebra associated to the filtration induced by the valuation is commutative. A complete description is given of all such valuations. Apart from the valuation induced by the Bernstein filtration itself, the rest are parameterized by a triple \((p,q,\alpha)\in\mathbb{Z}\setminus\mathbb{N}\times\mathbb{N}\setminus\{0\}\times k^*\) such that \(q\leq-p\). This description relies on a proposition to the effect that the valuations under consideration are completely determined by their restricitions to the subfield \(k(XY^{-1})\) of \(D_1(k)\).