Valuation ideals in polynomial rings

Abstract

Introduction. Among the valuation ideals in a polynomial ring = K [x, y] in two indeterminates, the ones of central importance are the simple valuation ideals, that is, the valuation ideals which are not products of two ideals different from 0, since every valuation ideal has a unique factorization into simple valuation ideals. The problem of the characterization of simple valuation ideals has been dealt with by Zariski, in the case that the field K is algebraically closed and of characteristic 0, in his paper, Polynomial ideals defined by infinitely near base points. There the problem is referred to the ring of holomorphic functions in x, y: *= K { x, y }, and a valuation ideal q in Z* is simple if and only if its general element is absolutely irreducible. If, however, only the characterization of the simple valuation ideals is desired, the notions of the general element of an ideal in D* and its absolute irreducibility are somewhat too strong for the problem set; although it should be stated that these notions are applied by Zariski to other topics not touched upon here. In this paper we treat the theory of simple valuation ideals by a more explicit and direct method and we also extend the theory to algebraically closed fields of arbitrary characteristic p 5 O. We characterize the simple v-ideals q in the sequence of zero-dimensional valuation ideals in ?, for a given valuation v, in terms of the value v(q) (under v) of q (that is, the least value assumed by elements of q) and of the least value greater than v(q) assumed by elements of ?). If q is not simple, an explicit factorization for q in terms of the two mentioned values is given Since in our treatment the field K is of arbitrary characteristic, Puiseux series expansions for valuations are not available. A corresponding tool is found in Theorem 6. There for a given valuation v, a certain (finite or infinite) sequence of polynomialsfi(x, y) is introduced. In the case that K is of characteristic 0, if v is given('), for example, by y=clxr(l)+c2xr(2)+ * * * , ciCK, r(i) =ri rational, with 0 <ri <ri+?, then the polynomials ft(x, y) correspond roughly to the irreducible polynomials gi(x, y), g2(x, y), * * which have y=clxr(l), y=clxr(l)+c2xr(2), I . * respectively as roots. We next reduce our considerations to valuations of rational rank 2. This reduction serves two purposes. First it unifies the discussion; but much more