Orderings, cuts and formal power series

Abstract

Each real closed field 'R' can be viewed as a subfield of the formal power series field κ((G)), where G and κ are respectively the value group and the residue field of the natural valuation v on R. One can prove that cuts in a given Hahn product H might be realized by suitable elements of a certain bigger Hahn product. In this way, each cut in R, and hence each ordering on R(y) corresponds to a canonically defined element Ψ in a certain bigger formal power series field κα ((Gα)) which contains κ((G)). These elements Ψ do not belong to R. More generally, one can prove that orderings on the ring R[y] are in one-to-one correspondence with canonically defined elements ϕ, where ϕ is an element Ψ as above or an element of R. To find ϕ, one first fixes an embedding ι of R into κ((G)), which is also proper, i.e., the value group of ι(R) is G. Then it can be seen that there exists a certain extension κα((Gα)) of κ((G)), so that ϕ ∈ κα ((Gα)). The correspondence between the orderings on the ring R[y] and the elements ϕ can be generalized to the case of the orderings on the ring R[y1,···,yn]. Actually, one can obtain an n-tuple (ϕ1,··· ,ϕn) corresponding to an ordering on R[y1···yn], where all the ϕi's belong to a certain Hahn product. If F is an ordered field having R as its real closure such that ι(F) ⊆ κ((V)), where V is the value group of F, then it can be proved that the value group W of F(ϕ1···ϕn) is generated over V by all the exponents ɣ appearing in some ϕi 1≤ i ≤ n. One can also find all the possible forms of W/V. Furthermore, it is possible to show that if an ordered abelian group extension W of V is given so that W/V has one of those forms, then there exists (ϕ1,....,ϕn) such that the value group of F (ϕ1,....ϕn) is W.