The exponential rank of nonarchimedean exponential fields

Abstract

Assume that w is a coarsening of a second valuation v, that is, R w ⊃ R v (in this case, we also say that v is finer than w). Then I w ⊂ I v , and if v is convex, it follows from condition 3) or 4) that also w is convex. There is always a finest convex valuation v, and the other convex valuations are precisely the coarsenings w of v. This valuation v is called the natural valuation of (K, <).Let us quickly recall how natural valuations are obtained already on a totally ordered abelian group (G, <). We set |a| := max{a, −a}. Two elements a, b are archimedean equivalent if there is some n ∈ N such that n|a| ≥ |b| and n|b| ≥ |a|. The equivalence class of a is called archimedean class of a and is denoted by [a].is the natural valuation of G. It satisfies the triangle inequality v(a + b) ≥ min{va, vb} and v(−a) = va as well as property (CO). The natural valuation of an ordered field (K, <) is just the natural valuation on its ordered additive group (K, +, 0, <). In this case, vK := v(K \{0}) forms a totally ordered abelian group endowed with the addition [a] + [b] := [ab], and v is a field valuation. It is characterized by the fact that its residue field is an archimedean ordered field. The valuation ideal I v is the set of all infinitesimals, and K >0 \ R v is the set of all positive infinite elements (elements which are incomparably bigger than 1). For more information on natural valuations, see [KS].Throughout this paper, K will be a nonarchimedean ordered field, and v will denote its nontrivial natural valuation. The natural valuations of the appearing ordered groups will be denoted by v G , and v G G shall be the ordered set v G (G \ {0}).By general valuation theory, the set R of all valuation rings R w of coarsenings w = v of v is totally ordered by inclusion, and it is order isomorphic to the set of all nontrivial convex subgroups of the value group vK (again ordered by inclusion). Its order type is called the rank of v (or in our case, of (K, <)); for convenience, we will identify it with R. The convex subgroup corresponding to R w isand the value group wK is canonically isomorphic to vK/G w . For example, the rank of an archimedean ordered field is empty since its natural valuation is trivial. The rank of the rational function field K = R(t) with any order is a singleton, R = {K}, and vK is the only nontrivial convex subgroup of the nontrivial archimedean value group vK. The reader may have noted that we are not using the classical definition of "rank" since we include the trivial valuation but exclude v. The present version will be more useful for our purposes.

Surreal Ordered Exponential Fields

“…Some of the methods employed in Sections 6–10 are adaptations or expansions of methods developed by Ressayre [40], van den Dries, Macintyre and Marker [47, 48], and D’Aquino, Knight, Kuhlmann and Lange [15] in their treatments of truncation closed embeddings into Hahn fields of ordered exponential fields and ordered fields with additional structure more generally. However, as is evident from Proposition 1.4, even in the case of an ordered field K the existence of a truncation closed embedding into a Hahn field does not suffice to establish the existence of an initial embedding into .…”

Valuation theory of exponential Hardy fields I

“…In view of (4), we can replace z by log z to get that wz w log z < 0. This proves (9). Now assume that wz = 0 and z > 0.…”

“…Now assume that wz = 0 and z > 0. If vz < 0, then by (9), vz < v log z < 0. If vz > 0, then vz −1 < 0 and by (9)…”

“…If R = R, then LE R,F (x) has exponential rank 1, in the sense of [9]. In general, exprk LE R,F (x) = exprk R + 1.…”

Differential equations over polynomially bounded o-minimal structures

“…Kuhlmann and Kuhlmann [11] for more information.) An exponential analog of Lemma 4 is easily obtained.…”