On the equivalence of types

Abstract

Types over a discrete valued field (K,v) are computational objects that pa- rameterize certain families of monic irreducible polynomials in Kv(x), where Kv is the completion of K at v. Two types are considered to be equivalent if they encode the same family of prime polynomials. In this paper, we characterize the equivalence of types in terms of certain data supported by them. based on the choice of certain key polynomials φi ∈ K(x) and positive rational numbers νi. Then, given an irreducible polynomial f ∈ K(x), he characterized all extensions of v to the field L := K(x)/(f) as limits of sequences of inductive valuations on K(x) whose value at f grows to infinity. In the case K = Q, Ore's p-regularity condition is satisfied when all valuations on L extending the p-adic valuation are sufficiently close to valuations onK(x) that may be obtained from µ0 by a single augmentation step. In 1999, J. Montes carried out Ore's program in its original formulation (3, 10). He introduced types as computational objects which are able to construct MacLane's valuations and the higher residual polynomial operators foreseen by Ore. These ideas made the whole theory constructive and well-suited to computational applications, and led to the design of several fast algorithms to perform arithmetic tasks in global fields (2, 4, 5, 7, 11). In 2007, M. Vaquie reviewed and generalized MacLane's work to non-discrete valuations. The introduction of the graded algebra Gr(µ) of a valuation µ led him to a more elegant presentation of the theory. In the papers (1) and (6), which deal only with discrete valuations, the ideas of Montes were used to develop a constructive treatment of Vaquie's approach, which included the computation of generators of the graded algebras and a thorough revision and simplification of the algorithmic applications.