arXiv:2002.10117v1 [math.LO] 24 Feb 2020 On Dimensions, Standard Part Maps, and p-Adically Closed Fields Ningyuan Yao Fudan University February 25, 2020 Abstract The aim of this paper is to study the dimensions and standard part maps between the field of p-adic numbers Qp and its elementary extension K in the language of rings Lr . We show that for any K-definable set X ⊆ K m , dimK (X) ≥ dimQp (X ∩ Qm p ). Let V ⊆ K be convex hull of K over Qp , and st : V → Qp be the standard part map. We show that for any K-definable function f : K m → K, there is m definable subset D ⊆ Qm p such that Qp \D has no interior, and for all x ∈ D, −1 either f (x) ∈ V and st(f (st (x))) is constant, or f (st−1 (x)) ∩ V = ∅. We also prove that dimK (X) ≥ dimQp (st(X ∩ V m )) for every definable X ⊆ m K . 1 Introduction In [15], L. van den Dries consider a pair (R, V ), where R is an o-minimal extension of a real closed field, and V is a convex hull of an elementary submodel M of R. Let µ ⊆ R be the set infinitesimals over M and V̂ = V /µ be the reside field with residue class map x 7→ x̂. If M is Dedekind complete in R, then V̂ = M and the residue class map coincide the standard part map st : R −→ M. In this context, van den Dries showed the follows: Theorem 1. [15] Let S ⊆ Rn be R-definable and Ŝ = {x̂| x ∈ S ∩ V n }. Then (i) S ∩ M n is definable in M and dimM (S ∩ M n ) ≤ dimR (S); (ii) st(S) is definable in M and dimM (st(S)) ≤ dimR (S). Theorem 2. [15] Let f : Rm → R be an R-definable function. Then here is a finite partition P of M m into definable sets, where each set in the partition is either open in M m or lacks of interior. On each open set C ∈ P we have: (i) either f (x) ∈ / V for all x ∈ C h ; (ii) or there is a continuous function g : C −→ M, definable in M, such that f (x) ∈ V and st(f (x)) = g(st(x)), for all x ∈ C h , where C h is the hull of C defined by h m C = {x̄ ∈ R |∃ȳ ∈ C m ^
(xi − yi ∈ µ) }. i=1 Remark 1.1. For any topological space Y , and X ⊆ Y , by Int(X) we mean the set of interiors in X. Namely, x ∈ Int(X) iff there is an open neighborhood B ⊆ Y of x contained in X. There are fairly good analogies between the field of reals R and the field of p-adic numbers Qp , in both model-theoretic and field-theoretic view. For example, both of them are complete and locally compact topological fields, are distal and dp-minimal structures, have quantifier eliminations with adding the new predicates for n-th power, and have cell decompositions. In this paper, we treat the p-adic analogue of above two Theorems, where M is replaced by Qp , and R is replaced by an arbitrary elementary extension K of Qp . In our case, the convex hull V is the set    x ∈ K| x = 0 ∨ ∃n ∈ Z v(x) > n and µ, the infinitesimals of K over Qp , is the set    x ∈ K| x = 0 ∨ ∀n ∈ Z v(x) > n . By Lemma 2.1 in [11], for every x ∈ V , there is a unique element st(x) in Qp such that a−st(a) ∈ µ, we call it the standard part of a and st : a 7→ st(a) the standard part map. It is easy to see that st : V −→ Qp is a surjective ring homomorphism and st−1 (0) = µ. So V̂ = V /µ is isomorphic to Qp in our context. With the notations as above, we now highlight our main results. Theorem 1.2. Let S ⊆ K n be K-definable. Then (i) S ∩ Qnp is definable in Qp and dimQp (S ∩ Qnp ) ≤ dimK (S); (ii) st(S ∩ V n ) is definable in Qp and dimQp (st(S ∩ V n )) ≤ dimK (S). Theorem 1.3. Let f : K m → K be an K-definable function. Then here is a finite partition P of Qp into definable sets, where each set in the partition is either open in Qm p or lacks of interior. On each open set C ∈ P we have: (i) either f (x) ∈ / V for all x ∈ C h ; (ii) or there is a continuous function g : C −→ Qp , definable in Qp , such that f (x) ∈ V and st(f (x)) = g(st(x)), for all x ∈ C h . In the rest of this introduction we give more notations and model-theoretic approach. 1.1 Notations Let p denote a fixed prime number, Qp the field the p-adic field, and v : Qp \{0} → Z is the valuation map. Let K be a fixed elementary extension of Qp . Then valuation v extends to a valuation map from K\{0} to ΓK , we also denote it by v, where (ΓK , +, < , 0) is the corresponding elementary extension of (Z, +, <, 0). Fact 1.4. Let v : K\{0} −→ ΓK be as above. Then we have • v(xy) = v(x) + v(y) for all x, y ∈ K; • v(x + y) ≥ min{v(x), v(y)}, and v(x + y) = min{v(x), v(y)} if v(x) 6= v(y); • For x ∈ Qp , |x| = p−v(x) if x 6= 0 and |x| = 0 if x = 0 defines a non-archimedean metric on Qp . • K is a non–archimedean topological field. We will assume a basic knowledge of model theory. Good references are [12] and [9]. We will be referring a lot to the comprehensive survey [1] for the basic model theory of the p-adics. A key point is Macintyres theorem [8] that Th(Qp , +, ×, 0, 1) has quantifier elimination in the language L = Lr ∪ {Pn | n ∈ N+ }, where Lr is the language of rings, and the predicate Pn is interpreted as the n-th powers {y ∈ Qp | y 6= 0 ∧ ∃x(y = xn )} for each n ∈ N+ . Note that Pn is definable in Lr . Moreover, the valuation is definable in Lr as follows. Fact 1.5. [3] Let f, g ∈ K[x1 , ..., xm ]. Then {ā ∈ K m | v(f (ā)) ≤ v(g(ā))} is definable. Remark 1.6. It is easy to see from Fact 1.5 that {a ∈ K|v(a) = γ} and {a ∈ K|v(a) < γ} are definable for any fixed γ ∈ ΓK . For A a subset of K, by an LA -formula we mean a formula with parameters from A. By x̄, ȳ, z̄ we mean arbitrary n-variables and ā, b̄, c̄ ∈ K n denote n-tuples in K n with n ∈ N+ . By |x̄|, we mean the length of the tuple x̄. We say that X ⊆ K m is A-definable if there is a LA -formula φ(x1 , ..., xm ) such that X = {(a1 , ..., am ) ∈ K m | K |= φ(a1 , ..., am )}. We also denote X by φ(K m ) and say that X is defined by φ(x̄). We say that X is definable in K if X ⊆ K m is K-definable. If X ∈ Qm p is defined by some LQp -formula m ψ(x̄). Then by X(K) we mean ψ(K ), namely, the realizations of ψ in K, which is a definable subset of K m . For any subset A of K, by acl(A) we mean the algebraic closure of A. Namely, b ∈ acl(A) if and only if there is a formula φ(x) with parameters from A such that b ∈ φ(K) and φ(K) is finite. Let α = (α1 , ..., αn ) ∈ K m , we denote acl(A ∪ {α1 , ..., αn }) by acl(A, α). By a saturated extension K of K, we mean that |K| is a sufficiently large cardinality, and every type over A ⊆ K is realized in K whenever |A| < |K|. 1.2 Preliminaries The p-adic field Qp is a complete, locally compact topological field, with basis given by the sets B(a,n) = {x ∈ Qp | x = a ∧ v(x − a) ≥ n} for a ∈ Qp and n ∈ Z. The elementary extension K ≻ Qp is also a topological field but not need to be complete or locally compact. Let X ⊆ K m , we say that ā ∈ X is an interior if there is γ ∈ Γk such that B(b̄,γ) = {(b1 , ..., bm ) ∈ K| n ^ v(ai − bi ) > γ} ⊆ X. i=1 Let V = {x ∈ K| x = 0 ∨ ∃n ∈ Z(v(x) > n)}. We call V the convex hull of Qp . It is easy to see that for any a, b ∈ K, if b ∈ V and v(a) > v(b), then a ∈ V . As we said before, for every a ∈ V , there is a unique a0 ∈ Qp such that v(a − a0 ) > n for all n ∈ Z. This gives a map a 7→ a0 from V onto Qp . We call this map the standard part map, denoted by st : V −→ Qp . For any ā = (a1 , ..., am ) ∈ V m , by st(ā) we mean (st(a1 ), ..., st(am )). Let f (x̄) ∈ K[x̄] be a polynomial with every coefficient contained in V . Then by st(f ), we mean the polynomial over Qp obtained by replace each coefficient of f by its standard part. Let µ = {a ∈ K| st(a) = 0}, which is the collection of all infinitesimals of K over Qp . It is easy to see that for any a ∈ K\{0}, a ∈ / V iff a−1 ∈ µ. Any definable subset X ⊆ K n has a topological dimension which is defined as follows: Definition 1.7. Let X ⊆ K n . By dimK (X), we mean the maximal k ≤ n such that the image of the projection π : X −→ K k ; (x1 , ..., xn ) 7→ (xr1 , ..., xrk ) has interiors, for suitable 1 ≤ r1 < ... < rk ≤ n. We call dimK (X) the topological dimension of X. Recall that Qp is a geometry structure (see Definition 2.1 and Proposition 2.11 of [7] ), so any K |= Th(Qp ) is a geometry structure. The fields has geometric structure are certain fields in which model-theoretic algebraic closure equals field-theoretic algebraic closure. Every geometry structure is a pregeometry structure, which means that for any ā = (a1 , ..., an ) ∈ K n and A ⊆ K, dim(ā/A) makes sense, which by definition is the maximal k such that ar1 ∈ / acl(A) and ari+1 ∈ / acl(A, ar1 , ..., ari ) for some subtuple (ar1 , ..., ark ) of ā. We call dim(ā/A) the algebraic dimension of ā over A. Fact 1.8. [7] Let A be a subset of K and X an A-definable subset of K m . (i) If ā ∈ K m and b̄ ∈ K n . Then we have dim(ā, b̄/A) = dim(ā/A, b̄) + dim(b̄/A) = dim(b̄, ā/A). (ii) Let K ≻ K be a saturated model. Then dimK (X) = max{dim(ā/A)| ā ∈ X(K)}. (iii) Let φ(x1 , ..., xm , y1 , ..., yn ) be any LA -formula and r ∈ N. Then the set {b̄ ∈ K n | dimK (φ(K m , b̄)) ≤ r} is A-definable. (iv) If X ⊆ K is K-definable. Then X is infinite iff dimK (X) ≥ 1. (v) Let A0 be a countable subset of Qp , and let Y be an A0 -definable subset of Qnp . Then there is ā0 ∈ Y such that dim(ā0 /A0 ) = dimQp (Y ). It is easy to see from Fact 1.8 that for any LK -formula φ(x1 , ..., xn ) and K ′ ≻ K, we have n dimK (φ(K n )) = dimK ′ (φ(K ′ )). We will write dimK (X) by dim(X) if there is no ambiguity. If the function f : X −→ K is definable in K, and Y ⊆ X × K is the graph of f . Then we conclude directly that dim(X) = dim(Y ) by Fact 1.8 (ii). For later use, we recall some well-known facts and terminology. Hensel’s Lemma. Let Zp = {x ∈ Qp |x = 0 ∨ v(x) ≥ 0} be the valuation ring of Qp . Let f (x) be a polynomial over Zp in one variable x, and let a ∈ Zp such that v(f (a)) > 2n + 1 and v(f ′ (a)) ≤ n, where f ′ denotes the derivative of f . Then there exists a unique â ∈ Zp such that f (â) = 0 and v(â − a) ≥ n + 1. We say a field E is a Henselian field if Hensel’s Lemma holds in E. Note that to be a henselian field is a first-order property of a field in the language of rings. Namely, there is a Lr -sentence σ such that E |= σ iff E is a henselian field. So any K ≻ Qp is henselian. 2 Main results 2.1 Some Properties of Henselian Fields Since Qp is complete and local compact, it is easy to see that: Fact 2.1. Suppose that E is a finite (or algebraic) field extension of Qp . Then for any α ∈ E\Qp , there is n ∈ Z such that v(α − a) < n (|α − a| > p−n ) for all a ∈ Qp . Namely, Qp is closed in E. We now show that Fact 2.1 holds for any K |= Th(Qp ). Lemma 2.2. Let K be a henselian field, R = {x ∈ K| x = 0 ∨ v(x) ≥ 0} be the valuation ring of K, and f (x) ∈ R[x] a polynomial, D ⊆ ΓK a cofinal subset, and X = {xd | d ∈ D} ⊆ R. If lim f (xd ) = 0, d∈D,d→+∞ Then there exist a cofinal subset I ⊆ D and a ∈ K such that lim xi = a and f (a) = 0. i∈I,i→+∞ Proof. Induction on deg(f ). Suppose that f has degree 1, say, f (x) = αx + β. Then for any γ ∈ ΓK , there is d0 ∈ D such that v(f (xd )) > γ for all d0 < d ∈ D. Now v(αxd + β) > γ implies that v(xd − (− αβ )) > γ − v(α). So β |xd − (− )| = 0 d∈D,d→+∞ α lim and hence xd = − lim d∈D,d→+∞ β β and f (− ) = 0 α α as required. Now suppose that deg(f ) = n + 1 > 1. We see that the derivative f ′ has degree n. If there are γ0 ∈ ΓK and ε0 ∈ D such that v(f ′ (xε )) ≤ γ0 for all ε0 < ε ∈ D. Take ε0 sufficiently large such that v(f (xε )) > 4γ0 + 1 for all ε0 < ε ∈ D. Then, by Hensel’s Lemma, we see that for all ε > ε0 , there is x̂ε such that v(f (xε )) − 1 v(x̂ε − xε ) ≥ and f (x̂ε ) = 0 2 As f has at most finitely many roots, there is a cofinal subset I ⊆ D and some x̂ε ∈ K such that v(f (xi )) − 1 v(x̂ε − xi ) > 2 for all i ∈ I. Since v(f (xi )) → +∞, we see that v(x̂ε − xi ) → +∞. Thus we have lim xi = x̂ε and f (x̂ε ) = 0, i∈I,i→+∞ as required. Otherwise, if for every γ ∈ ΓK , there is γ < dγ ∈ D such that v(f ′(xdγ )) > γ. Then there is a cofinal subset I = {dγ | γ ∈ ΓK } ⊆ D such that lim f ′ (xi ) = 0, i∈I,i→+∞ Then, by induction hypothesis, there exist a cofinal subset J ⊆ I and b ∈ K such that lim xj = b and f ′ (b) = 0. j∈J,j→+∞ Since f is continuous, limj∈J,j→+∞f (xj ) = f (b). Now J is cofinal in I, and I is cofinal in D, we conclude that J is cofinal in D. This complete the proof. Proposition 2.3. If K is a henselian field, and E is a finite extension of K. Then for any α ∈ E\K, there is γ0 ∈ ΓK such that v(α − a) < γ0 for all a ∈ K. Namely, K is closed in E. Proof. By ([5], Lemma 4.1.1), the valuation of K extends uniquely to E. For each β ∈ E, Let g(x) = xn + an−1 xn−1 + ... + a1 x + a0 be the minimal polynomial of β over K, then the valuation of β is exactly v(an0 ) (See [6], Prop. 5.3.4). Let α ∈ E\K, and d(x) be the minimal polynomial of α over K with degree k. Then d(x + a) is the minimal polynomial of (α − a) over K for any a ∈ K. Since . We d(x + a) = xf (x) + d(a) for some f (x) ∈ K[x], we see that v(α − a) = v(d(a)) k claim that there is γ0 ∈ ΓK such that v(d(a)) < γ0 for all a ∈ K. Otherwise, we will find a sequence {aγ | γ ∈ ΓK } such that v(d(aγ )) > γ. Replace d(x) by ǫd(x) with some ǫ sufficiently close to 0, we may assume that d ∈ R[x]. Moreover, fix γ0 ∈ Γ, if v(α − a) > γ0 , and v(α − b) > 2γ0, then v(a − b) ≥ γ0 . So {b ∈ K| v(α − b) > γ0 } ⊆ δ0 R for some δ0 ∈ K, and hence {b ∈ K| v(d(b)) > γ0 } = {b ∈ K| kv(α − b) > γ0 } ⊆ kδ0 R. Let δ = kδ0 . If δ ∈ R, then, by Lemma 2.2, there is b ∈ K such that d(b) = 0. However d is minimal polynomial of degree > 1, so has no roots in K. A contradiction. If δ ∈ / R, then δ −1 ∈ R. Suppose that d(x) = dk xk + ... + d1 x + d0 . Let h(x) = dk xk + ... + d1 δ −k+1 x + d0 δ −k . We see that h(x) ∈ R[x] and h(δ −1 x) = dk (δ −1 x)k + ... + d1 δ −k+1 (δ −1 x) + d0 δ −k = δ −k d(x). Now we have v(h(δ −1 aγ )) = v(δ −k d(aγ )) > γ − kv(δ). For γ > γ0 , we have aγ ∈ δR. Therefore δ −1 aγ ∈ R for all γ > γ0 . Applying Lemma 2.2 to h(x), we can find c ∈ K such that h(c) = h(δ −1 δc) = 0 = δ −k d(δc). So d(δc) = 0. A contradiction. Now we assume that K is an elementary extension of Qp in the language of rings Lr . This follow result was proven by [13] in the case of K = Qp . P Lemma 2.4. Let x̄ = (x1 , ..., xm ) and f (x̄, y) = ni=0 pi (x̄)y i ∈ K[x̄, y]. Then there is a partition of n _ m R = {x̄ ∈ K | pi (x̄) 6= 0 ∧ ∃y(f (x̄, y) = 0) } i=0 into finitely many definable subsets S, over each of which f has some fixed number k ≥ 1 of distinct roots in K with fixed multiplicities m1 , ..., mk . For any fixed x̄0 ∈ S, let the roots of f (x̄0 , y) be r1 , ..., rk , and e = max{v(ri − rj )| 1 ≤ i < j ≤ k}. Then x̄0 has a neighborhood N ⊆ K m , γ ∈ ΓK , and continuous, definable functions F1 , ..., Fk : S ∩ N −→ K such that for each x̄ ∈ S ∩ N, F1 (x̄), ..., Fk (x̄) are roots of f (x̄, y) of multiplicities m1 , ..., mk and v(Fi (x̄) − ri ) > 2e. Proof. The proof of Lemma 1.1 in [13] applies almost word for word to the present context. The only problem is that the authors used Fact 2.1 in their proof. But the Proposition 2.3 saying that we could replace Qp by arbitrary K |= Th(Qp ) in our argument. Remark 2.5. Lemma 1.1 of [13] saying that definable functions F1 , ..., Fk are not only continuous but analytic. However we can’t proof it in arbitrary K |= Th(Qp ) as K might not be complete as a topological field. Similarly, Lemma 1.3 in [13] could be generalized to arbitrary K |= Th(Qp ) as follows: Lemma 2.6. If A ⊆ K m and f : A −→ K is definable. Then there is a definable set B ⊆ A, open in K m such that A\B has no interior and f is continuous on B. Proof. The proof of Lemma 1.3 in [13] applies almost word for word to the present context. 2.2 Dimensions We now assume that K is an elementary extension of Qp . Lemma 2.7. Suppose that A ⊆ K, X, Y are A-definable in K, f : X −→ Y is an A-definable function. If f is a finite-to-one map, dim(X) = dim(f (X)). Proof. Let K be a saturated elementary extension of K. By Fact 1.8 (iii), there is r ∈ N such that |f −1 (y)| ≤ r for all y ∈ Y (K). For any a ∈ X(K), since |{b ∈ X| f (b) = f (a)}| ≤ r, we see that a ∈ acl(A, f (a)). So dim(a/A, f (a)) = 0. By Fact 1.8 (i) we have dim(a/A) = dim(a, f (a)/A) = dim(a/A, f (a)) + dim(f (a)/A) = 0 + dim(f (a)/A). So dim(a/A) = dim(f (a)/A). By Fact1.8 (ii), we conclude that dim(X) = dim(Y ). Lemma 2.8. Suppose that A ⊆ K, f : X −→ Y is an A-definable function in K. Then dim(X) ≥ dim(f (X)). Proof. Generally, we have dim(a/A) = dim(a, f (a)/A) = dim(a/A, f (a)) + dim(f (a)/A) ≥ dim(f (a)/A). By Fact1.8 (ii), we conclude that dim(X) ≥ dim(Y ). Corollary 2.9. Suppose that A ⊆ K, f : X −→ Y is an A-definable bijection function in K. Then dim(X) = dim(f (X)). Proof. f −1 is a definable function as f is bijection. So we conclude that dim(X) ≥ dim(f (X)) = dim(Y ) ≥ dim(f −1 (Y )) = dim(X). Lemma 2.10. Suppose that X, Y are A-definable in K. Then dim(X ∪ Y ) = max{dim(X), dim(Y )}. Proof. By Fact1.8 (ii). Lemma 2.11. Let X ⊆ K n . Then dim(X) is the minimal k ≤ n such that there is definable Y ⊆ X with dim(Y ) = dim(X) and projection π : X −→ K k ; (x1 , ..., xn ) 7→ (xr1 , ..., xrk ) is a finite-to-one map on Y , for suitable 1 ≤ r1 < ... < rk ≤ n. Proof. Let k be as above and π : X −→ K k be a projection with π(x1 , ..., xn ) = (xr1 , ..., xrk ). If Y ⊆ X such that the restriction π ↾ Y : Y −→ K k is a finite-to-one map. Then by Lemma 2.7 we have dim(Y ) = dim(π(Y )) and hence dim(X) = dim(Y ) = dim(π(Y )) ≤ k. Now suppose that dim(X) = l ≤ k. Without loss of generality, we assume that f : X −→ K l ; (x1 , ..., xn ) 7→ (x1 , ..., xl ) is a projection such that f (X) has nonempty interior. We claim that Claim. Let Z0 = {b ∈ K l | f −1 (b) is finite } and Z1 = K l \Z0 . Then dim(Z1 ) < l. Proof. Clearly, Z1 = {b ∈ K l | dim(f −1 (b)) ≥ 1} is definable in K. If dim(Z1 ) = l. Then, there is β ∈ Z1 (K) such that dim(β/A) = l, where is K ≻ K is saturated. Since dim(f −1 (β)) ≥ 1, by Fact 1.8 (ii), there is α ∈ dim(f −1 (β)) such that dim(α/A, β) ≥ 1. By Fact 1.8 (i), we conclude that dim(α/A) = dim(α, f (α)/A) = dim(a/A, f (α)) + dim(f (α)/A) ≥ l + 1. But dim(α/A) ≤ dim(X) = l. A contradiction. Since dim(Z1 ) < l, by Lemma 2.10, dim(Z0 ) = l. The restriction of f on f −1 (Z0 ) is a finite-to-one map, we conclude that dim(f −1 (Z0 )) = dim(Z0 ) = l = dim(X) by Lemma 2.7. Now dim(f −1 (Z0 )) = dim(X) and the restriction of f on f −1 (Z0 ) is a finite-to-one map. So k ≤ l as k is minimal. We conclude that k = l = dim(X) as required. Corollary 2.12. Let X ⊆ K n be definable with dim(X) = k. Then there exists a partition of X into finitely many K-definable subsets S such that whenever dim(S) = dim(X), there is a projection πS : S −→ K k on k suitable coordinate axes which is finite-to-one. Proof. Let X0 = X and [n]k be the set of all subset of {1, ..., n} of cardinality k. By Lemma 2.11, there exist D0 = {r1 , ..., rk } ∈ [n]k , S0 ⊆ X with dim(S0 ) = dim(X0 ), such that the projection π : (x1 , ..., xn ) 7→ (xr1 , ..., xrk ) is finite-to-one on S0 and infinite-to-one on X0 \S0 . If dim(X0 \S0 ) < dim(X0 ), then the partition {X0 \S0 , S0 } meets our requirements. Otherwise, let X1 = X0 \S0 , we could find D1 ∈ [n]k \{D0 } and S1 ∈ X1 such that the projection on coordinate axes from D1 is finite-to-one over S1 . Repeating the above steps, we obtained sequences Xi and Si such that Xi+1 = Xi \Si . As [n]k is finite, there is a minimal t ∈ N such that dim(Xt ) < dim(X0 ) and dim(Si ) = dim(X0 ) for all i < t. It is easy to see that {S0 , ..., St−1 , Xt } meets our requirements. Recall that by [14], Th(Qp ) admits definable Skolem functions. Namely, we have Fact 2.13. [14] Let A ⊆ K and φ(x̄, y) be a LA -formula such that K |= ∀x̄∃yφ(x̄, y). Then there A-definable function f : K m → K such that K |= ∀x̄φ(x̄, f (x̄)). With the above Fact, we could refine Corollary 2.12 as follows: Corollary 2.14. Let X ⊆ K n be definable with dim(X) = k. Then there exists a partition of X into finitely many K-definable subsets S such that whenever dim(S) = dim(X), there is a projection πS : S −→ K k on k suitable coordinate axes which is injective. Proof. Let X0 = X. By Corollary 2.12, we may assume that the projection π : X0 −→ K k given by (x1 , ..., xn ) 7→ (x1 , ..., xk ) is finite-to-one. By compactness, there is r ∈ N such that |π −1 (ȳ) ∩ X0 | ≤ r for all ȳ ∈ K k . Induction on r. If r = 1, then π is injective on X0 . Otherwise, by Fact 2.13, there is a definable function f : π(X) −→ X such that π(f (ȳ)) = x for all ȳ ∈ π(X0 ). It is easy to see that f is injective and hence, by Corollary 2.7, S0 = f (π(X0 )) is a definable subset of X of dimension k. Moreover π : S0 −→ K k is exactly the inverse of f , hence injective. If dim(X0 \S0 ) then the partition {X0 \S0 , S1 } satisfies our require. Otherwise, X1 = X0 \S0 has dimension k and |π −1 (ȳ) ∩ X1 | ≤ r − 1 By our induction hypothesis, there is a partition of X1 into finitely may definable subsets meets our requirements. This completes the proof. Theorem 2.15. Let B ⊆ K m be definable in K. Then dimK (B) ≥ dimQp (B ∩ Qm p ). Proof. Suppose that dimK (B) = k. By Lemma 2.10 and Corollary 2.14, we may assume that π : B −→ K k is injective. The restriction of π to B ∩ Qm p is a injective projection m m k from B ∩ Qp to Qp . By Lemma 2.11, dimQp (B ∩ Qp ) ≤ k. Note that Pn (K) = {a ∈ K| a 6= 0 ∧ ∃b ∈ K(a = bn )} is an open subset of K whenever K is a hensilian field. For any polynomial f (x1 , ..., xm ) ∈ K[x1 , ..., xm ], Pn (f (K m )) = {a ∈ K m | f (a) 6= 0 ∧ ∃b ∈ K(f (a) = bn )} is an open subset of K m since f is continuous. 2.3 Standard Part Map and Definable Functions The following Facts will be used later. Fact 2.16. [2] Every complete n-type over Qp is definable. Equivalently, for any K ≻ Qp , any Lr -formula φ(x1 , ..., xn , y1 , ..., ym), and any b̄ ∈ K m , the set {ā ∈ Qnp | K |= φ(ā, b̄)} is definable in Qp . Fact 2.17. [10] Let X ⊆ K m be a Qp -definable open set, let Y ⊆ X be a K-definable subset of X. Then either Y or X\Y contains a Qp -definable open set. Fact 2.18. [10] Let X ⊆ K m be a K-definable set. Then st(X) ∩ st(K m \X) has no interior. Recall that µ is the collection of all infinitesimals of K over Qp , which induces a equivalence relation ∽µ on K, which is defined by a ∽µ b ⇐⇒ a − b ∈ µ. Definition 2.19. Let f (x̄, y), g(x̄, y) ∈ K[x̄, y] be polynomials. By f ∽µ g we mean that P P (i) if |x̄| = 0, f (y) = ni=1 ai y i , and g(y) = ni=1 bi y i , then f ∽µ g iff ai ∽µ bi for each i ≤ n. P P (ii) if |x̄| > 0, f (x̄, y) = ni=1 ai (x̄)y i , and g(y) = ni=1 bi (x̄)y i , where a’s and b’s are polynomials with variables from x̄. Then f ∽µ g iff ai ∽µ bi for each i ≤ n. Lemma 2.20. Let x̄ = (x1 , ..., xn ), f (x̄) and g(x̄) be polynomials over K with f ∽µ g. If ā = (a1 , ..., an ) and b̄ = (b1 , ..., bn ) are tuples from K with ai ∽µ bi for each i ≤ n. Then f (ā) ∽µ g(b̄). Proof. We see that α ∈ µ iff v(α) > Z. Since v(α + β) ≥ min{v(α), v(β)} and v(αβ) = v(α) + v(β), we see that µ is closed under addition and multiplication. As polynomials are functions obtained by compositions of addition and multiplication, we conclude that f (ā) ∽µ g(b̄). Since V ⊆ K is also closed under addition and multiplication. We conclude directly that: Corollary 2.21. Let x̄ = (x1 , ..., xn ), and f (x̄) ∈ K[x̄] be a polynomial with every coefficient contained in V . If a = (a1 , ..., an ) ∈ V n , then st(f )(st(a)) = st(f (a)). Corollary 2.22. Let f (x) = an xn + ... + a1 x1 + a0 be a polynomial over K with every coefficient contained in V and an ∈ / µ. If b ∈ K such that f (b) = 0. Then b ∈ V . Proof. Suppose for a contradiction that b ∈ / V . Then st(b−1 ) = 0. Clearly, we have b−n f (b) = an + ... + a1 b−n+1 + a0 b−n = 0. Let g(y) = a0 y n + ... + an−1 y + an . Then g(b−1 ) = 0. By Corollary 2.21, we have st(g)(st(b−1 )) = 0. As st(b−1 ) = 0, we see that st(an ) = 0, which contradicts to an ∈ / µ. Fact 2.23. [10] Let S ⊆ K m be definable in K. Then st(S ∩ V m ) ⊆ Qm p is definable in Qp . Lemma 2.24. Let f : K k −→ K be definable in K. Then (i) X∞ = {a ∈ Qkp | f (a) ∈ / V } is definable in Qp . (ii) Let X = Qkp \X∞ . Then g : X −→ Qp given by a 7→ st(f (a)) is definable in Qp Proof. By Fact 2.16, there is a LQp -formula φ(x, y) such that for all a ∈ Qkp and b ∈ Qp , we have Qp |= φ(a, b) ⇐⇒ v(f (a)) < v(b). Hence a ∈ X∞ ⇐⇒ Qp |= ∀yφ(a, y), which shows that X∞ is definable in Qp . Again by Fact 2.16, there is LQp -formula ψ(x, y1 , y2 ) such that for all a ∈ Qkp , b1 , b2 ∈ Qp , M |= ψ(a, b1 , b2 ) ⇐⇒ v(f (a) − b1 ) > v(b2 − b1 ). Therefore b = st(f (a)) ⇐⇒ Qp |= ∀y1 ∀y2 (v(b − y1 ) > v(y1 − y2 ) → ψ(a, y1 , y2 )). for all a ∈ Qkp and b ∈ Qp . We conclude that g : X −→ Qp , a 7→ st(f (a)) is definable in Qp m m Lemma 2.25. Let X ⊆ Qm and st(ā) ∈ / X, then p be a clopen subset of Qp . If ā ∈ V ā ∈ / X(K). Proof. As X is clopen and st(ā) ∈ / X, there is N ∈ Z such that B(ā,N ) = {b̄ ∈ Qm p | m ^ v(bi − st(ai )) > N} ∩ X = ∅. i=1 So B(ā,N ) (K) ∩ X(K) = ∅. But v(ai − st(ai )) > Z, hence m ā ∈ {b̄ ∈ K | m ^ v(bi − st(ai )) > N} = B(ā,N ) (K). i+1 So ā ∈ / X(K) as required. Lemma 2.26. If X ⊆ K m and f : X −→ K are definable in K, then there is a polynomial q(x1 , ..., xm , y) such that the graph of f is contained in the variety {(a1 , ..., am , b) ∈ K m+1 | q(a1 , ..., am , b) = 0}. Proof. Let Y be W the graph of f . Since Th(Qp ) has quantifier elimination, Y is defined by a disjunction si=1 φi (x̄), where each φi (x̄) is a conjunction ( li ^ j=1 gij (x̄, y) = 0) ∧ li ^ Pnij (hij (x̄, y)), j=1 where g’s and h’s belong to K[x̄, y]. Now each Pnij (hij (x̄, y)) defines an open subset of K m+1 . Since dim(Y ) ≤ m, we see that for each i ≤ s, there is f (i) ≤ li such that gif (i) 6= 0 . Let q(x̄, y) = Πsi=1 gif (i) (x̄, y). Then Y ⊆ {(a1 , ..., am , b) ∈ K m+1 | q(a1 , ..., am , b) = 0} as required. Proposition 2.27. If f : K m −→ K is definable in K. Let X = {ā ∈ Qm p | f (ā) ∈ V }. Then   −1 DX = {ā ∈ X| ∃b̄, c̄ ∈ st (ā) f (b̄) − f (c̄) ∈ / µ }. has no interiors. Proof. By Lemma 2.26, there is a polynomial g(x1 , ..., xm , y) ∈ K[x1 , ..., xm , y] such that the graph of f is contained in the variety of g. Without loss of generality, we may assume that each coefficient of g is in V , otherwise, we could replace g by g/c, where c is a coefficient of g with minimal valuation. Moreover, we could assume that at least one coefficient of g is not in µ. Suppose for a contradiction that DX contains a open subset of Qm p . Shrink DX if necessary, we may assume that DX ⊆ X is a Qp -definable open set in Qm p . By Lemma m 2.4, there is a partition P of DX (K) ⊆ K into finitely many definable subsets S, over each of which g has some fixed number k ≥ 1 of distinct roots in K with fixed multiplicities m1 , ..., mk . For any fixed x̄0 ∈ S, let the roots of g(x̄0 , y) be r1 , ..., rk , and e = max{v(ri − rj )| 1 ≤ i < j ≤ k}. Then x̄0 has a neighborhood N ⊆ K m , γ ∈ ΓK , and continuous, definable functions F1 , ..., Fk : S ∩ N −→ K such that for each x̄ ∈ S ∩ N, F1 (x̄), ..., Fk (x̄) are roots of g(x̄, y) of multiplicities m1 , ..., mk and v(Fi (x̄) − ri ) > 2e. Since DX (K) is a Qp -definable open subset of X(K). By Fact 2.17, some S ∈ P contains a Qp -definable open subset ψ(K m ) of X(K). Where ψ is an LQp -formula. Let m A0 = φ(Qm p ). Then A0 ⊆ A is an open subset of Qp , and over A0 (K) we have (i) g has some fixed number k ≥ 1 of distinct roots in K with fixed multiplicities m1 , ..., mk . (ii) For any fixed x̄0 ∈ A0 (K), let the roots of g(x̄0 , y) be r1 , ..., rk , and e = max{v(ri − rj )| 1 ≤ i < j ≤ k}. Then x̄0 has a neighborhood N ⊆ K m , γ ∈ ΓK , and continuous, definable functions F1 , ..., Fk : A0 (K) ∩ N −→ K such that for each x̄ ∈ A0 (K) ∩ N, F1 (x̄), ..., Fk (x̄) are roots of g(x̄, y) of multiplicities m1 , ..., mk and v(Fi (x̄) − ri ) > 2e. (iii) for any ā ∈ A0 , there exist b̄, c̄ ∈ st−1 (ā) such that st(f (b̄)) 6= st(f (c̄)). Suppose that g(x1 , ..., xm , y) = n X gi (x1 , ..., xm )y i , i=0 where each gi (x̄) ∈ K[x̄]. Since the variety {ā ∈ Qm p | st(gn )(ā) = 0} has dimension m − 1, it has no interior. By Fact 2.17, A0 \{ā ∈ Qm p | st(gn )(ā) = 0} contains a open subset of Qm p . Without loss of generality, we may assume that {ā ∈ Qm p | st(gn )(ā) = 0} ∩ A0 = ∅. Since the family of clopen subsets forms a base for topology on Qm p , we may assume that A0 is clopen. We now claim that Claim 1. For every ā ∈ A0 (K), gn (ā) ∈ / µ. Proof. Otherwise, by Corollary 2.21, we have st(gn )(st(ā)) = st(gn (ā)) = 0. So st(ā) ∈ / A0 . By Lemma 2.25, we see that ā ∈ / A0 (K). A contradiction. By Claim 1 and Corollary 2.22, we see that for every ā ∈ A0 (K) and b ∈ K, if g(ā, b) = 0, then b ∈ V . By Corollary 2.21, we conclude the following claim Claim 2. For every ā ∈ A0 (K) and b ∈ K, if g(ā, b) = 0, then b ∈ V and st(g)(st(ā), st(b)) = 0. Now st(g) is a polynomial over Qp . Applying Lemma 2.4 to st(g) and Fact 2.17, and shrink A0 if necessary, we may assume that • st(g) has some fixed number d ≥ 1 of distinct roots in K with fixed multiplicities n1 , ..., nd over A0 . • Fix some x̄0 ∈ A0 , let the roots of st(g)(x̄0 , y) (in Qp ) be s1 , ..., sd , and ∆ = max{v(si − sj )| 1 ≤ i < j ≤ d}. Then there are definable continuous functions H1 , ..., Hd : A0 −→ Qp such that for each x̄ ∈ A0 , H1 (x̄), ..., Hd (x̄) are roots of st(g)(x̄, y) of multiplicities n1 , ..., nd and v(Hi (x̄) − si ) > 2∆. • for any ā ∈ A0 , there exist b̄, c̄ ∈ st−1 (ā) such that st(f (b̄)) 6= st(f (c̄)). By Claim 2, we see that for any x̄ ∈ A0 (K), and b ∈ K, if g(x̄, b) = 0, then b ∽µ Hi (st(x̄)) for some i ≤ d. As g(x̄, f (x̄)) = 0 for all x̄ ∈ K m , we see that Claim 3. For each x̄ ∈ A0 (K), f (x̄) ∽µ Hi (st(x̄)) for some i ≤ d. Let Di = {x̄ ∈ A(K)| v(f (x̄) − si ) > 2∆}. We claim that S Claim 4. A0 (K) = di=1 Di and Di ∩ Dj = ∅ for each i 6= j. Namely, {D1 , ..., Dd } is a partition of A0 (K). Proof. Let x̄ ∈ A0 (K). By Claim 3, there is some i ≤ d such that f (x̄) ∽µ Hi (st(x̄)). It is easy to see that v(f (x̄) − si ) = v(f (x̄) − Hi (st(x̄)) + Hi (st(x̄)) − si ) = v(Hi (st(x̄)) − si ) > 2∆. S So x̄ ∈ Di and this implies that A0 (K) = di=1 Di . On the other side, if x̄ ∈ Di ∩ Dj for some 1 ≤ i < j ≤ d, we have v(f (x̄) − si ) > 2∆ and v(f (x̄) − sj ) > 2∆, which implies that v(si − sj ) = v(si − f (x̄) + f (x̄) − sj ) ≥ min{v(si − f (x̄)), v(f (x̄) − sj )} ≥ 2∆. But v(si − sj ) ≤ ∆. A contradiction. Claim 5. Let i ≤ d. For any ā, b̄ ∈ Di , if ā ∽µ b̄ then st(f (ā)) = st(f (b̄)). Proof. Let x̄ ∈ Di . By Claim 3, there is j ≤ d such that st(f (x̄)) = Hj (st(x̄)). We see that v(f (x̄) − sj ) = v(f (x̄) − Hj (st(x̄)) + Hj st(x̄)) − sj ) = v(Hj (st(x̄)) − sj ) > 2∆. So x̄ ∈ Dj . By Claim 4, i = j. We conclude that st(f (x̄)) = Hi (st(x̄)) whenever x̄ ∈ Di . This complete the proof of Claim 5. Recall that for any ā ∈ A0 , there exist b̄, c̄ ∈ st−1 (ā) such that st(f (b̄)) 6= st(f (c̄)). By Claim 4 and Claim 5, we see that for each ā ∈ A0 , there is 1 ≤ i 6= j ≤ d such that ā ∈ st(Di ) ∩ st(Dj ). This means that [ A0 ⊆ st(Di ) ∩ st(Dj ) 1≤i6=j≤d By Fact 2.18, each st(Di ) ∩ st(Dj ) has no interior. By Fact 2.17, A0 has no interiors. A contradiction. Corollary 2.28. If f : K m −→ K is definable in K. Let X∞ = {ā ∈ Qm / V }. p | f (ā) ∈ Then   −1 U = {ā ∈ X∞ | ∃b̄, c̄ ∈ st (ā) f (b̄) ∈ V ∧ f (c̄) ∈ / V }. has no interior. Proof. Otherwise, suppose that U ⊆ K m is open. Applying Proposition 2.27 to g(x) = (f (x))−1 , we see that g(U) ⊆ V , and for all ā ∈ U there are b̄, c̄ ∈ st−1 (a) such that st(g(b̄)) 6= 0 and st(g(c̄)) = 0. A contradiction. Lemma 2.29. Let f : K k −→ K be definable in K, X = {a ∈ Qkp | f (a) ∈ V }, and X∞ = {a ∈ Qkp | f (a) ∈ / V }. Then both   −1 DX = {ā ∈ X| ∃b̄, c̄ ∈ st (ā) f (b̄) − f (c̄) ∈ / µ }. and   U = {ā ∈ X∞ | ∃b̄, c̄ ∈ st (ā) f (b̄) ∈ V ∧ f (c̄) ∈ / V }. −1 are definable sets over Qp Proof. Let X0 = {ā ∈ X| st(f (ā)) = 0} and X1 = {ā ∈ X| st(f (ā)) 6= 0}. As we showed in Lemma 2.24, both X0 and X1 are Qp -definable sets. Let g : K k \f −1 (0) −→ K be the K-definable function given by x̄ 7→ 1/f (x̄). Let Y ⊆ K k+1 be the graph of f and Z ⊆ K k+1 be the graph of g. For each ā ∈ Qkp , let st(Y )ā = {b ∈ Qp | (ā, b) ∈ st(Y )} and st(Z)ā = {b ∈ Qp | (ā, b) ∈ st(Z)}. Let S1 = {ā ∈ X| | st(Y )ā | > 1}, S2 = {ā ∈ X0 | | st(Z)ā | ≥ 1}, and S3 = {ā ∈ X1 | | st(Z)ā | > 1}. We now show that DX = S1 ∪ S2 ∪ S3 . Clearly, S1 and S3 are subsets of DX . If ā ∈ S2 , then st(f (ā)) = 0 and there is b̄ ∈ st−1 (ā) such that 1/f (b̄) ∈ V , so f (ā) − f (b̄) ∈ / µ, which implies that ā ∈ DX . Therefore, we conclude that S1 ∪ S2 ∪ S3 ⊆ DX . Conversely, take any ā ∈ DX and suppose that b̄, c̄ ∈ st−1 (ā) such that f (b̄) − f (c̄) ∈ / µ. If both f (b̄) and f (c̄) are in V , then ā ∈ S1 ; If f (b̄) ∈ / V and st(f (ā)) = 0, then b̄ ∈ dom(g). We see that (ā, 0) ∈ st(Z), so ā ∈ S2 ; If f (b̄) ∈ / V and st(f (ā)) 6= 0, then ā, b̄ ∈ dom(g), st(g(b̄)) = 0 and st(g(ā)) 6= 0, which implies that | st(Z)ā | > 1, and thus ā ∈ S3 . So we conclude that DX ⊆ S1 ∪ S2 ∪ S3 as required. As S1 , S2 , and S3 are definable sets over Qp , DX is definable over Qp . Similarly, U is definable over Qp . h Suppose that C ⊆ Qm p , we define the hull C by C h = {x̄ ∈ K m | st(x̄) ∈ C}. Theorem 2.30. Let f : K m → K be an K-definable function. Then here is a finite partition P of Qp into definable sets, where each set in the partition is either open in Qm p or lacks of interior. On each open set C ∈ P we have: (i) either f (x) ∈ / V for all x ∈ C h ; (ii) or there is a continuous function g : C −→ Qp , definable in Qp , such that f (x) ∈ V and st(f (x)) = g(st(x)), for all x ∈ C h . Proof. Let X, X∞ be as in Lemma 2.24, DX as in Proposition 2.27, and U as in Corollary 2.28, then DX and U have no interior, and by Lemma 2.29, they are definable. Now {DX , X\DX , U, X∞ \U} is a partition of Qm p . Clearly, {Int(X∞ \U), (X∞ \U) \ Int(X∞ \U)} is a partition of X∞ \U where Int(X∞ \U) is open and (X∞ \U) \ Int(X∞ \U) lacks of interior. Let h : X\DX −→ Qp be a definable function defined by x 7→ st(f (x)). By Theorem 1.1 of [13], there is a finite partition P ∗ of X\DX into definable sets, on each of which h is analytic. Each set in the partition is either open in Qm p or lacks of interior. Clearly, the partition P = {DX , U, Int(X∞ \U), (X∞ \U) \ Int(X∞ \U)} ∪ P ∗ satisfies our condition. We now prove our last result. Lemma 2.31. Let Z ⊆ K n be definable in K of dimension k < n, and the projection π : (x1 , ...xn ) 7→ (x1 , ..., xk ) is injective on Z. Then dimQp (st(Z ∩ V n )) ≤ k. Proof. As π is injective on X, there is a definable function f = (f1 , ..., fn ) : K k −→ K n such that • f (π(x̄)) = (f1 (π(x̄)), ..., fn (π(x̄))) = x̄ for all x̄ ∈ Z; • f (ȳ) = (0, ..., 0) for all ȳ ∈ K k \π(X). By Lemma 2.26, for each i ≤ n, there is a polynomial Fi (ȳ, u) such that the graph of fi is contained in the variety V (Fi ) = {(ȳ, u) ∈ K k+1 | Fi (ȳ, u) = 0} of Fi . We assume that each coefficient belongs to V . It is easy to see that for each (a1 , ..., an ) ∈ Z ∩ V n , we have fi (π(a1 , ..., an )) = ai . So Fi (a1 , ..., ak , ai ) = 0. By Corollary 2.21, st(Fi )(st(a1 ), ..., st(ak ), st(ai ))) = 0. So st(Z ∩ V n ) is contained in the variety V (st(F1 ), ..., st(Fn )) =  (a1 , ..., an ) ∈ Qnp | n  ^ st(Fi )(st(a1 ), ..., st(ak ), st(ai ))) = 0 i=1  . Let A ⊆ Qp be the collection of all coefficients from st(Fi )’s. Then for each (a1 , ..., an ) ∈ V (st(F1 ), ..., st(Fn )), we see that ai is a root of Fi (a1 , ..., ak , u), and hence ai ∈ acl(A, a1 , ..., ak ), where i ≤ n. This implies that dim(a1 , ..., an /A) = dim(a1 , ..., ak /A) ≤ k for all (a1 , ..., an ) ∈ V (st(F1 ), ..., st(Fn )). By Fact 1.8 (v), we see that  dimQp (V (st(F1 ), ..., st(Fn ))) = max dim(a1 , ..., an /A)| (a1 , ..., an ) ∈ V (st(F1 ), ..., st(Fn )) ≤ k. So st(Z ∩ V n ) ≤ k as required. Theorem 2.32. Let Z ⊆ K n be definable in K. Then dimQp (st(Z ∩ V n )) ≤ dimK (Z). Proof. 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