Forking and Dividing in Fields with Several Orderings and Valuations
نویسنده
چکیده
variety). For each i, let φi(x) be a C-dense quantifier-free Li-formula with parameters from K. Then we can find a K-definable rational function f : C → P which is non-constant, and has the property that the divisor f−1(0) is a sum of distinct points in ⋂n i=1 φi(K), with no multipliticities. (In particular, the support of the divisor contains no points from C(K)\C(K) and no points from C \ C.) Proof. Let g be the genus of C. Claim 4.4. We can find g + 1 distinct points p1, . . . , pg+1 in ⋂n i=1 φi(K) ⊂ C(K). Proof. For each i, choose a model Ki of Ti extending K Li. Then φi(Ki) is Zariski dense in C(K i ). This (easily) implies that φi(Ki) g+1 is Zariski dense in Cg+1(K i ). If U denotes the subset of C g+1 consisting of (x1, . . . , xg+1) such that xi 6= xj for every i and j, then U is a Zariski dense Zariski open subset of C, because its complement is a closed subvariety of lower dimension. The intersection of a Zariski dense set with a Zariski dense Zariski open is still Zariski dense. So φi(Ki) g+1 ∩ U is still Zariski dense in C. Let ψi(x1, . . . , xg+1) be the following quantifier-free Li-formula over K:
منابع مشابه
∗−orderings and ∗−valuations on Algebras of Finite Gelfand-kirillov Dimension
Considerable work has been done in developing the relationship between ∗-orderings, ∗valuations and the reduced theory of Hermitian forms over a skewfield with involution [12] [13] [14] [15] [16] [23] [24]. This generalizes the well-known theory in the commutative case; e.g., see [4] [6] [7] [27]. In the commutative theory, formally real function fields provide a rich source of examples [6]. In...
متن کاملAn Independence Theorem for Ntp2 Theories
We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (whic...
متن کاملForking and dividing in NTP₂ theories
We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded forking assuming NTP2.
متن کاملThorn-forking as Local Forking
We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, fu...
متن کامل