Fields Institute Communications Volume 00, 0000

A Density Property for PpC-fields

Joachim Schmid

Universität Konstanz Fachbereich Mathematik und Statistik 78457 Konstanz Germany

                             schmidj@fmi.uni-konstanz.de

Abstract. We prove an approximation and density result for the set of rational points of an absolutely irreducible variety defined over a padically closed fields.

Introduction

In 1984 Heinemann and Prestel [3] introduced a class of fields which satisfy a local global principle for rational points on absolutely irreducible varieties referring to a finite number of local closures (i.e. real closures or henselizations) of the field. They proved [3, Theorem 1.9] that a field K belonging to this class satisfies a certain approximation condition, namely for an absolutely irreducible K-variety there exists a K-rational point simultaneously close to a given sequence of regular points of V in the local closures.

Their result includes the special case of pseudo real closed fields and pseudo padically closed fields with only finitely many orderings or p-valuations respectively.

In [7] it was shown that the approximation condition of Heinemann and Prestel holds for arbitrary pseudo real closed fields and it is the aim of this paper to prove that it also holds for arbitrary pseudo p-adically closed fields.

1 Preliminaries

We assume that the reader is familiar with the theory of p-valuations and the notion of a pseudo p-adically closed field. We just fix some notations.

Let p ∈ ℕ be a fixed prime and suppose that K is a field with char(K) = 0. A valuation v : K ⟶ Γ ∪ {∞} is called a p-valuation if v(p) is the minimal positive element of v(K) and the residue class field of K w.r.t. v is isomorphic to 𝔽_{p}. The valuation ring of a p-valuation is called a p-valuation ring.

If O is a p-valuation ring of the field K, then two p-adic closures (L_{1}, O_{1}) and (L_{2}, O_{2}) of (K, O) are isomorphic over (K, O) if and only if K ∩ L_{1}^{n} = K ∩ L_{2}^{n} for

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

2000 Mathematics Subject Classification. Primary 12 J 12; secondary 12 J 10.

all natural numbers n (cf. [5, Cor. 3.11]). For arbitrary p-valued field this is not the case in general. However, if K is pseudo p-adically closed then two p-adic closures of (K, O) are isomorphic over (K, O) (cf. [2, Prop. 3.06]). In this case we denote the p-adic closure of (K, O) by \overline{(K, O)}.

The p-adic Kochen operator we denote by γ_{p}(X), i.e.

γ_{p}(X) = \frac{1}{p} · \frac{X^{p} − X}{(X^{p} − X)^{2} − 1}.

By X_{K}^{p} we denote the set of all p-valuation rings of the field K. The set X_{K}^{p} is endowed with the so-called Harrison topology. A subbasis for the open sets of this topology consists of the sets H(a) := {O ∈ X_{K}^{p} | a ∈ O} (a ∈ K \ {0}). One can easily see that H(a) ∪ H(b) = H(a^{2} + pb^{2}) and X_{K}^{p} \ H(a) = H((pa)^{−1}). Therefore the sets H(a) even form a basis for the topology and moreover they are clopen. It is well-known that X_{K}^{p} is a compact Hausdorff space w.r.t. the Harrison topology.

If v is a valuation on K and a = (a_{1}, …, a_{n}) ∈ K^{n} we let v(a) := min{v(a_{1}), …, v(a_{n})}.

First we prove an elementary lemma.

Lemma 1.1 Let (K, v) be a p-valued field and suppose that u_{1}, …, u_{n}, u ∈ K. Let further t := ∑_{i=1}^{n} p^{i−1}u_{i}^{n+1} + p^{n}u^{n+1}. Then

v(u) < min{v(u_{1}), …, v(u_{n})} ⟺ v(t) ≥ v(p^{n}u^{n+1}).

Proof Assume first that v(u) < min{v(u_{1}), …, v(u_{n})}. As v(p) is the minimal positive element of the value group of v we obtain v(u_{i}) ≥ v(p) + v(u). Thus for all i ∈ {1, …, n} we get

v(p^{i−1}u_{i}^{n+1}) = (i − 1) · v(p) + (n + 1) · v(u_{i}) ≥ (n + i) · v(p) + (n + 1) · v(u)
= i · v(p) + v(p^{n}u^{n+1}) > v(p^{n}u^{n+1})

and hence even v(t) = v(p^{n}u^{n+1}). Now suppose that v(t) ≥ v(p^{n}u^{n+1}). Then

v(∑_{i=1}^{n} p^{i−1}u_{i}^{n+1}) = v(t − p^{n}u^{n+1}) ≥ v(p^{n}u^{n+1}).

As v(p) is the minimal positive element of the value group all the summands p^{i−1}u_{i}^{n+1} have different values and thus

v(∑_{i=1}^{n} p^{i−1}u_{i}^{n+1}) = min{v(p^{i−1}u_{i}^{n+1}) | 1 ≤ i ≤ n}.

Hence v(p^{i−1}u_{i}^{n+1}) ≥ v(p^{n}u^{n+1}) for all i. Thus v(u_{i}^{n+1}) ≥ (n + 1 − i) · v(p) + v(u^{n+1}) > v(u^{n+1}). This finally yields v(u_{i}) > v(u) for i ∈ {1, …, n} and the lemma is proved.

Lemma 1.2 Let K be a formally p-adic field and let a_{1}, …, a_{n} ∈ K. Then there is an element a ∈ K such that

v(a) = min{v(a_{1}), …, v(a_{n})}

for all p-valuations v of K.

Proof Let R be the Kochen ring of K. As R is a Bezout domain with K as its field of quotients (cf. [5, Theorem 6.17]) the fractional ideal (a_{1}, …, a_{n}) generated by a_{1}, …, a_{n} is principal. Thus there is an element a ∈ K such that (a) = (a_{1}, …, a_{n}). Obviously a has the desired property.

The real counterpart of the following proposition is an easy consequence of the fact that the theory RCF of real closed fields admits elimination of quantifiers. However, the theory pCF of p-adically closed fields is only model complete and does not admit elimination of quantifiers in the language of valued fields. Therefore the proof requires some more careful arguments. We adopt an idea which implicitly can be found in [1] and [2].

Proposition 1.3 Let K be a pseudo p-adically closed field and suppose that φ is a first order formula in the language of valued fields with parameters from the field K. Then the set \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ φ\} is clopen.

Proof As \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ φ\} is the complement of \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ¬φ\} it suffices to show that all the sets \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ φ\} are open.

Since the theory pCF of p-adically closed fields is model complete there is an existential formula χ with parameters from K such that

pCF ⊧ (φ ⟷ χ).

χ is of the form ∃x_{1}, …, x_{n} ψ for some quantifier-free formula ψ. We may assume that ψ is in disjunctive normal form, i.e., ψ = ⋁_{j=1}^{m} ψ_{j} and each formula ψ_{j} is a conjunction of atomic formulas and negations of atomic formulas. Then

\{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ φ\} = ⋃_{j=1}^{m}\{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}\}

and hence it suffices to prove that all the sets \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}\} are open.

Now fix j ∈ {1, …, m}. The formula ψ_{j} is of the form

⋀_{i=1}^{k} f_{i}(x) = 0 ∧ ⋀_{i=1}^{ℓ} g_{i}(x) ≠ 0 ∧ ⋀_{i=1}^{r} p_{i}(x) ∈ O ∧ ⋀_{i=1}^{s} q_{i}(x) ∉ O

for certain polynomials f_{1}, …, f_{k}, g_{1}, …, g_{ℓ}, p_{1}, …, p_{r}, q_{1}, …, q_{s} ∈ K[X_{1}, …, X_{n}].

We first show that w. l. o. g. we may assume that ℓ = r = s = 0. By the usual Rabinowitsch trick we may remove all the inequalities g_{i}(x) ≠ 0. Obviously

pCF ⊧ (q_{i}(x) ∉ O ⟷ ∃y(y ∈ O ∧ pyq_{i}(x) = 1))

and we also may assume that s = 0. Finally if L is a p-adically closed field then {γ_{p}(a) | a ∈ L} is the unique p-valuation ring of L. Hence

pCF ⊧ (p_{i}(x) ∈ O ⟷ ∃y p · ((y^{p} − y)^{2} − 1) · p_{i}(x) = y^{p} − y)

and we also may remove the conditions p_{i}(x) ∈ O.

In order to show that \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}\} is open let O ∈ X_{K}^{p} and suppose \overline{(K, O)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}. Hence there are a_{1}, …, a_{n} ∈ \overline{(K, O)} such that \overline{(K, O)} ⊧ ψ_{j}(a_{1}, …, a_{n}), i.e., f_{i}(a_{1}, …, a_{n}) = 0 (1 ≤ i ≤ k).

By the Primitive Element Theorem there is some a ∈ \overline{(K, O)} such that K(a_{1}, …, a_{n}) = K(a). We may assume that a is integral over O. Let h(X) = ∑_{i=0}^{d−1} c_{i}X^{i} + X^{d} be the minimal polynomial of a over K. Then h ∈ O[X] as O is integrally closed in K. Furthermore h′(a) ≠ 0 since char(K) = 0. Each a_{i} is of the form a_{i} = h_{i}(a) for certain h_{1}, …, h_{n} ∈ K[X]. Thus f_{i}(h_{1}(a), …, h_{n}(a)) = f_{i}(a_{1}, …, a_{n}) = 0 and hence there are polynomials F_{1}, …, F_{k} ∈ K[X] such that

f_{i}(h_{1}(X), …, h_{n}(X)) = F_{i}(X) · h(X) (1 ≤ i ≤ k).

Since K is dense in \overline{(K, O)} (cf. [2, Prop. 3.05]) and h(a) = 0, h′(a) ≠ 0 by continuity of polynomials there is an element b ∈ K such that v(b) ≥ 0, h′(b) ≠ 0 and v(h(b)) > 2v(h′(b)). Then

U := H(b, c_{0}, …, c_{d} − 1, \frac{h(b)}{ph′(b)^{2}} )

is an open neighbourhood of O. We claim that U ⊂ \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}}. If O′ ∈ U then b ∈ O′, h ∈ O′[X] and v′(h(b)) > 2v′(h′(b)). Hence by the Hensel-Rychlik Lemma there is some z ∈ \overline{(K, O′)} such that h(z) = 0 and thus f_{i}(h_{1}(z), …, h_{n}(z)) = 0 (1 ≤ i ≤ k). Therefore \overline{(K, O′)} ⊧ ∃x_{1}, …, x_{n} ψ_{j}. This proves Proposition 1.3.

2 Approximation and density results

If K is a pseudo p-adically closed field then it is well-known that two different p-valuation rings induce different topologies on K. Theorem 2.1 is a generalization of this result.

Theorem 2.1 Let K be a pseudo p-adically closed field and suppose that X_{K}^{p} = · ⋃_{i=1}^{m}U_{i} is a partition of X_{K}^{p} into clopen sets. Let further a_{1}, …, a_{m} ∈ K and b_{1}, …, b_{m} ∈ K^{×}. Then there is an element a ∈ K such that

v(a − a_{1}) ≥ v(b_{1})  if O_{v} ∈ U_{1}
⋮
v(a − a_{m}) ≥ v(b_{m})  if O_{v} ∈ U_{m}

for all p-valuations v of K.

Proof The clopen sets U_{1}, …, U_{m} are of the form U_{i} = H(c_{i}) for certain c_{1}, …, c_{m} ∈ K^{×}. We let

f_{i}(X) := (X − a_{i})^{2} + pb_{i}^{2}c_{i}^{2}  (1 ≤ i ≤ m)
g_{i}(X) := ∏_{j≠i} f_{j}(X)  (1 ≤ i ≤ m)

and define further g(X) := ∏_{i=1}^{m} f_{i}(X) and h(X) := ∑_{i=1}^{m} b_{i}^{2}g_{i}(X). An elementary computation shows

\frac{h(X)}{g(X)} = ∑_{i=1}^{m} \frac{b_{i}^{2}}{f_{i}(X)}  (1)

Further we define the rational function

F (X, Y, Z) := γ_{p}(Y) + γ_{p}(Z) − \frac{g(X)}{h(X)} ∈ K(X, Y, Z).  (2)

Then

F (X, Y, Z) = \frac{f(X, Y, Z)}{h(X) · q(Y) · q(Z)}  (3)

where q is the denominator of the Kochen operator, i.e., q(X) := p·((X^{p} −X)^{2} −1) and

f(X, Y, Z) = h(X) · (Y^{p} − Y) · q(Z) + h(X) · q(Y) · (Z^{p} − Z) − g(X) · q(Y) · q(Z).

By [6, Cor. 1] the polynomial f is absolutely irreducible. An easy computation shows

\frac{∂f}{∂Z} (X, Y, 0) = −h(X) · q(Y)  (4)

Now we show that V_{reg}(f) ∩ \overline{(K, O)}^{3} ≠ ∅ for all p-valuation rings O of K.

Let O ∈ U_{j} = H(c_{j}) and let v the p-valuation corresponding to O. Then f_{j}(a_{j}) = pb_{j}^{2}c_{j}^{2} ≠ 0 and v(f_{j}(a_{j})) = v(p) + v(b_{j}^{2}) + 2v(c_{j}) > v(b_{j}^{2}). Hence

v(\frac{b_{j}^{2}}{f_{j}(a_{j})}) < 0.  (5)

If i ≠ j then f_{i}(a_{j}) = (a_{j} − a_{i})^{2} + pb_{i}^{2}c_{i}^{2} ≠ 0 and

v(f_{i}(a_{j})) = v((a_{j} − a_{i})^{2} + pb_{i}^{2}c_{i}^{2})
= min{v((a_{j} − a_{i})^{2}), v(pb_{i}^{2}c_{i}^{2})}
≤ v(p) + v(b_{i}^{2}) + 2v(c_{i}).

As v(p) is the minimal positive element and v(c_{i}) < 0 we obtain v(p) + 2v(c_{i}) < 0 and hence v(f_{i}(a_{j})) < v(b_{i}^{2}). Therefore

v(\frac{b_{i}^{2}}{f_{i}(a_{j})}) > 0  for all i ≠ j.  (6)

Thus g(a_{j}) = ∏_{i=1}^{m} f_{i}(a_{j}) ≠ 0 and from (1), (5) and (6) we obtain

v(\frac{h(a_{j})}{g(a_{j})}) < 0.

Hence h(a_{j}) ≠ 0 and

v(\frac{g(a_{j})}{h(a_{j})}) > 0.

Therefore there is an element y ∈ \overline{(K, O)} such that q(y) ≠ 0 and

\frac{g(a_{j})}{h(a_{j})} = γ_{p}(y).

Now (2), (3) and (4) imply f(a_{j}, y, 0) = 0 and

\frac{∂f}{∂Z} (a_{j}, y, 0) = −h(a_{j}) · q(y) ≠ 0.

Thus (a_{j}, y, 0) is a regular point of the hypersurface f = 0.

Since K is a pseudo p-adically closed field and g(X), h(X), q(Y), q(Z) ∉ I(V(f)) = (f) there is a regular rational point (a, b, c) ∈ V_{reg}(f) ∩ K^{3} such that g(a), h(a), q(b), q(c) ≠ 0. Hence F (a, b, c) = 0 and thus (2) yields

\frac{g(a)}{h(a)} = γ_{p}(b) + γ_{p}(c).

Therefore by (1)

v(∑_{i=1}^{m} \frac{b_{i}^{2}}{f_{i}(a)}) = v(\frac{h(a)}{g(a)}) = v((γ_{p}(b) + γ_{p}(c))^{−1}) ≤ 0  (7)

for all p-valuations v of K. Now we show that a has the desired approximation properties. Let O ∈ U_{j} = H(c_{j}) and denote the corresponding valuation by v. Then for i ≠ j we obtain

v(f_{i}(a)) = v((a − a_{i})^{2} + pb_{i}^{2}c_{i}^{2})
= min{v((a − a_{i})^{2}), v(pb_{i}^{2}c_{i}^{2})}
≤ v(p) + v(b_{i}^{2}) + 2v(c_{i}).

As v(p) is the minimal positive element and v(c_{i}) < 0 we obtain v(p) + 2v(c_{i}) < 0 and hence v(f_{i}(a)) < v(b_{i}^{2}). This yields

v(\frac{b_{i}^{2}}{f_{i}(a)}) > 0  for alli ≠ j

and from (7) we get

v(\frac{b_{j}^{2}}{f_{j}(a)}) ≤ 0.

Therefore

v(b_{j}^{2}) ≤ v(f_{j}(a))
= v((a − a_{j})^{2} + pb_{j}^{2}c_{j}^{2})
= min{v((a − a_{j})^{2}), v(pb_{j}^{2}c_{j}^{2})}
≤ v((a − a_{j})^{2}).

Hence v(a − a_{j}) ≥ v(b_{j}) and the theorem is proved.

Corollary 2.2 Let K be a pseudo p-adically closed field and suppose that X_{K}^{p} = · ⋃_{i=1}^{m}U_{i} is a partition of X_{K}^{p} into clopen sets. Let further a_{1}, …, a_{m} ∈ K^{×}. Then there is an element a ∈ K such that

v(a) = v(a_{1})  if O_{v} ∈ U_{1}
⋮
v(a) = v(a_{m})  if O_{v} ∈ U_{m}

for all p-valuations v of K.

Proof By Theorem 2.1 there is an element a ∈ K such that v(a − a_{i}) ≥ v(pa_{i}) > v(a_{i}) for all p-valuations v such that O_{v} ∈ U_{i} (1 ≤ i ≤ m). Hence

v(a) = v((a − a_{i}) + a_{i}) = min{v(a − a_{i}), v(a_{i})} = v(a_{i})

whenever O_{v} ∈ U_{i}.

Theorem 2.3 (Block approximation) Let K be a pseudo p-adically closed field and suppose that V ⊂ K̃^{n} is an absolutely irreducible K-variety. Let further a_{1}, …, a_{m} ∈ K^{n}, b_{1}, …, b_{m} ∈ K^{×} and suppose that X_{K}^{p} = · ⋃_{i=0}^{m}U_{i} is a partition of X_{K}^{p} into pairwise disjoint clopen sets. Assume that V_{reg} ∩ \overline{(K, O)}^{n} ≠ ∅ for O ∈ U_{0} and suppose that for each i ∈ {1, …, m} and each O ∈ U_{i} there is a regular point x ∈ V_{reg} ∩ \overline{(K, O)}^{n} such that v(x−a_{i}) > v(b_{i}) where v is the valuation corresponding to O. Then there is a regular rational point a ∈ V_{reg} ∩ K^{n} such that

v(a − a_{1}) > v(b_{1})  if O_{v} ∈ U_{1}
⋮
v(a − a_{m}) > v(b_{m})  if O_{v} ∈ U_{m}

for all p-valuations v of K.

Proof By the assumption V has a regular point in each p-adic closure of K. Thus V admits a regular K-rational point a_{0}. Let b_{0} := 1. By Lemma 1.2 there is an element c ∈ K^{×} such that

v(c) = min({v(b_{0}), …, v(b_{m})} ∪ {v(a_{i} − a_{j}) | 0 ≤ i, j ≤ m})

for all p-valuations v of K and by Corollary 2.2 there are d_{0}, …, d_{m} ∈ K^{×} such that for all p-valuations v of K holds

v(d_{i}) = v(b_{i})  if O_{v} ∈ U_{i}
v(d_{i}) = v(p^{−1}c)  if O_{v} ∉ U_{i}

for 0 ≤ i ≤ m. We let a_{ij} be the coordinates of a_{0}, …, a_{m}, i.e., a_{i} = (a_{i1}, …, a_{in}) and define

h_{i}(X_{1}, …, X_{n}) := ∑_{j=1}^{n} p^{j−1}(X_{j} − a_{ij})^{n+1} + p^{n}d_{i}^{n+1} (0 ≤ i ≤ m).

Further we define the rational function

F (X, Y_{1}, Y_{2}, Y_{3}) := γ_{p}(Y_{1}) + γ_{p}(Y_{2}) + γ_{p}(Y_{3}) − X.

Then

F (X, Y_{1}, Y_{2}, Y_{3}) = \frac{f(X, Y_{1}, Y_{2}, Y_{3})}{q(Y_{1}) · q(Y_{2}) · q(Y_{3})}

where q(X) := p · ((X^{p} − X)^{2} − 1) again is the denominator of the p-adic Kochen operator and

\begin{aligned}f(X, Y_{1}, Y_{2}, Y_{3}) := & (Y_{1}^{p} − Y_{1})q(Y_{2})q(Y_{3}) + (Y_{2}^{p} − Y_{2})q(Y_{1})q(Y_{3}) \\ & + (Y_{3}^{p} − Y_{3})q(Y_{1})q(Y_{2}) − Xq(Y_{1})q(Y_{2})q(Y_{3}). \end{aligned}

An easy computation yields

\frac{∂f}{∂Y_{3}} (X, Y_{1}, 0, 0) = p · q(Y_{1}).

We further define

g_{i}(X_{1}, …, X_{n}, Y_{1i}, Y_{2i}, Y_{3i}) :=

f(p^{−n}d_{i}^{−(n+1)}h_{i}(X_{1}, …, X_{n}), Y_{1i}, Y_{2i}, Y_{3i}) (0 ≤ i ≤ m).

Let I(V) = (f_{1}, …, f_{r}) ⊂ K̃[X_{1}, …, X_{n}] be the vanishing ideal of V and define

W_{k} := V (f_{1}, …, f_{r}, g_{0}, …, g_{k}) ⊂ K̃^{n+3k+3} (0 ≤ k ≤ m).

Let further W := W_{m}. For every u ∈ K̃ the polynomial f(u, Y_{1}, Y_{2}, Y_{3}) ∈ K̃[Y_{1}, Y_{2}, Y_{3}] is the numerator of the rational function F (u, Y_{1}, Y_{2}, Y_{3}) ∈ K̃(Y_{1}, Y_{2}, Y_{3}). Thus by [6, Cor. 1] the polynomial f(u, Y_{1}, Y_{2}, Y_{3}) is irreducible. Furthermore f contains the monomial −p^{2}Y_{1}. Now a successive application of Theorem A.3 (i) of the appendix yields that W_{0}, …, W_{m} are absolutely irreducible K-varieties. In particular W is an absolutely irreducible K-variety. We show that W has a regular point in each p-adic closure of K.

Let O ∈ U_{i} for some i ∈ {0, …, m} and denote the corresponding valuation by v. Then there is a regular point x ∈ V_{reg} ∩ \overline{(K, O)}^{n} such that

v(x − a_{i}) > v(b_{i}) = v(d_{i}).

Note that one can choose x = a_{0} in the case i = 0. By Lemma 1.1 we obtain

v(h_{i}(x)) ≥ v(p^{n}d_{i}^{n+1})

and thus

p^{−n}d_{i}^{−(n+1)}h_{i}(x) = γ_{p}(ξ_{i})

for some ξ_{i} ∈ \overline{(K, O)} such that q(ξ_{i}) ≠ 0. Hence g_{i}(x, ξ_{i}, 0, 0) = 0 and \frac{∂g_{i}}{∂Y_{3i}} (x, ξ_{i}, 0, 0) = p · q(ξ_{i}) ≠ 0.

For j ≠ i we obtain

v(x − a_{j}) ≥ min{v(x − a_{i}), v(a_{i} − a_{j})} ≥ v(c) = v(pd_{j}) > v(d_{j})

and hence a further application of Lemma 1.1 yields

v(h_{j}(x)) ≥ v(p^{n}d_{j}^{n+1})

and thus

p^{−n}d_{j}^{−(n+1)}h_{j}(x) = γ_{p}(ξ_{j})

for some ξ_{j} ∈ \overline{(K, O)} such that q(ξ_{j}) ≠ 0. Hence g_{j}(x, ξ_{j}, 0, 0) = 0 and \frac{∂g_{j}}{∂Y_{3j}} (x, ξ_{j}, 0, 0) = p · q(ξ_{j}) ≠ 0.

If we let y_{k} := (ξ_{0}, 0, 0, ξ_{1}, 0, 0, …, ξ_{k}, 0, 0) (0 ≤ k ≤ m) then a successive application of Theorem A.3 (iv) yields that (x, y_{k}) is a regular point of W_{k}. In particular (x, y_{m}) ∈ W_{reg}.

Thus W has a regular point in each p-adic closure of K and q(Y_{ij}) ∉ I(W). As K is pseudo p-adically closed there is a regular rational point (a, z) ∈ W_{reg} ∩ K^{n+3m+3} such that q(z_{ij}) ≠ 0 for 0 ≤ i ≤ m, j = 1, 2, 3. Then a ∈ V_{reg} ∩ K^{n} by Theorem A.3(iii).

Finally we show that a satisfies the stated approximation conditions. Fix i ∈ {1, …, m}, let O ∈ U_{i} and denote the corresponding valuation by v. As

0 = g_{i}(a, z) = f(p^{−n}d_{i}^{−(n+1)}h_{i}(a), z_{1i}, z_{2i}, z_{3i})

and q(z_{1i}), q(z_{2i}), q(z_{3i}) ≠ 0 we get

p^{−n}d_{i}^{−(n+1)}h_{i}(a) = γ_{p}(z_{1i}) + γ_{p}(z_{2i}) + γ_{p}(z_{3i}).

Thus v(h_{i}(a)) ≥ v(p^{n}d_{i}^{n+1}) and hence we obtain from Lemma 1.1

v(a − a_{i}) > v(d_{i}) = v(b_{i})

and the theorem is proved.

If K is a pseudo p-adically closed field and O ∈ X_{K}^{p} we denote the unique extension of O to the p-adic closure \overline{(K, O)} of (K, O) by Ō.

Corollary 2.4 Let K be a pseudo p-adically closed field with Kochen ring R and suppose that V ⊂ K̃^{n} is an absolutely irreducible K-variety. Assume that for all O ∈ X_{K}^{p} there is a regular point x ∈ V_{reg} ∩ Ō^{n}. Then V_{reg} ∩ R^{n} ≠ ∅.

Proof Let U_{0} = ∅, U_{1} = X_{K}^{p}, a_{1} = (0, …, 0) and b_{1} = p^{−1} and apply Theorem 2.3.

Theorem 2.5 (Approximation Theorem) Let K be a pseudo p-adically closed field and suppose that V ⊂ K̃^{n} is an absolutely irreducible K-variety. Assume that O_{1}, …, O_{m} ∈ X_{K}^{p} are pairwise different p-valuation rings of the field K with corresponding p-valuations v_{i} : K ↠ Γ_{i} ∪ {∞} (1 ≤ i ≤ m). Suppose that V_{reg} ∩ \overline{(K, O)}^{n} ≠ ∅ for each p-valuation ring O of K and let x_{i} ∈ V_{reg} ∩ \overline{(K, O_{i})}^{n} (1 ≤ i ≤ m). Then for all α_{i} ∈ Γ_{i} (1 ≤ i ≤ m) there is a regular rational point a ∈ V_{reg} ∩ K^{n} such that v_{i}(a − x_{i}) > α_{i} (1 ≤ i ≤ m).

Proof Since K is pseudo p-adically closed it is dense in each of its p-adic closures (cf. [2, Prop. 3.05]). Thus there are a_{1}, …, a_{m} ∈ K^{n} such that v_{i}(x_{i}−a_{i}) > α_{i}. We choose b_{1}, …, b_{m} ∈ K^{×} such that v_{i}(b_{i}) = α_{i} (1 ≤ i ≤ m). For 1 ≤ i ≤ m we define

Y_{i} := \{O ∈ X_{K}^{p} | \overline{(K, O)} ⊧ ∃x(x ∈ V_{reg} ∧ v(x − a_{i}) > v(b_{i}))\}.

Obviously O_{i} ∈ Y_{i} and by Proposition 1.3 the sets Y_{i} are clopen. As O_{1}, …, O_{m} are pairwise different there are pairwise disjoint clopen subsets X_{1}, …, X_{m} of X_{K}^{p} such that O_{i} ∈ X_{i} (1 ≤ i ≤ m). Thus we obtain a partition

X_{K}^{p} = Z ·∪ · ⋃_{i=1}^{m}(X_{i} ∩ Y_{i})

for some clopen set Z ⊂ X_{K}^{p}. Now Theorem 2.3 yields a regular point a ∈ V_{reg} ∩K^{n} such that v(a − a_{i}) > v(b_{i}) for all p-valuations v with O_{v} ∈ X_{i} ∩ Y_{i}. In particular v_{i}(a − a_{i}) > v_{i}(b_{i}) = α_{i} and therefore

v_{i}(a − x_{i}) ≥ min{v_{i}(a − a_{i}), v_{i}(a_{i} − x_{i})} > α_{i} (1 ≤ i ≤ m).

Finally we restate Theorem 2.5 in terms of a density property.

Theorem 2.6 (Density Theorem) Let K be a pseudo p-adically closed field and suppose that V ⊂ K̃^{n} is an absolutely irreducible K-variety. Then the image of the canonical diagonal map

V_{reg} ∩ K^{n} ⟶ ∏_{O∈X_{K}^{p}} V_{reg} ∩ \overline{(K, O)}^{n}

is dense w.r.t. the product topology.

Proof Let U ⊂ ∏_{O∈X_{K}^{p}} V_{reg} ∩ \overline{(K, O)}^{n} be a non-empty open subset, say

(x_{O})_{O∈X_{K}^{p}} ∈ U. Then there is a family of open subsets U_{O} ⊂ V_{reg} ∩ \overline{(K, O)}^{n}, (O ∈ X_{K}^{p}) such that x_{O} ∈ U_{O}, ∏_{O∈X_{K}^{p}} U_{O} ⊂ U and U_{O} ≠ V_{reg} ∩ \overline{(K, O)}^{n} for only finitely many p-valuation rings, say O_{1}, …, O_{m}. Let v_{1}, …, v_{m} be the corresponding p-valuations. Since the p-adic closures of K are algebraic extensions of K there are α_{i} ∈ v_{i}(K) (1 ≤ i ≤ m) such that \{x ∈ V_{reg} ∩ \overline{(K, O_{i})}^{n} | v_{i}(x − x_{O_{i}}) > α_{i}\} ⊂ U_{O_{i}} (1 ≤ i ≤ m). Now Theorem 2.5 yields the result.

Appendix

In this appendix we will prove a theorem which already was used in the previous section in order to show that a certain algebraic set is an absolutely irreducible variety. The proof uses a classical result due to Dedekind and Mertens. The author is grateful to J. Ohm and A. Prestel for drawing his attention to this result.

If R is a commutative ring with 1 and f ∈ R[X_{1}, …, X_{n}] we let C_{R}(f) be the content ideal of f, i.e., the ideal generated by the coefficients of f in R.

Lemma A.1 (Dedekind-Mertens) Let R be a commutative ring with 1 and let f, g ∈ R[X_{1}, …, X_{n}]. Then

C_{R}(f)^{m+1} · C_{R}(g) = C_{R}(f)^{m} · C_{R}(fg)

for all sufficiently large m ∈ ℕ.

Proof Cf. [4, Lemma 2].

As a consequence we obtain

Proposition A.2 Let R be an integral domain with K as its field of quotients. Suppose that f ∈ R[X_{1}, …, X_{n}] is irreducible in K[X_{1}, …, X_{n}] and assume that C_{R}(f) = R. Then f generates a prime ideal in R[X_{1}, …, X_{n}].

Proof Since f is irreducible in K[X_{1}, …, X_{n}] the ideal fK[X_{1}, …, X_{n}] is prime. Therefore it suffices to show that

fK[X_{1}, …, X_{n}] ∩ R[X_{1}, …, X_{n}] = fR[X_{1}, …, X_{n}].

In order to show the non-trivial inclusion let h ∈ fK[X_{1}, …, X_{n}] ∩ R[X_{1}, …, X_{n}]. Hence h = fg for some g ∈ K[X_{1}, …, X_{n}].

We choose a ∈ R \ {0} such that ag ∈ R[X_{1}, …, X_{n}]. As C_{R}(f) = R we obtain from Lemma A.1

C_{R}(ag) = C_{R}(f · ag) = C_{R}(ah) ⊂ (a).

Therefore all the coefficients of ag are divisible by a and hence g ∈ R[X_{1}, …, X_{n}].

Theorem A.3 Let K be a field and suppose that V ⊂ K̃^{n} is an absolutely irreducible K-variety with vanishing ideal I(V) = (f_{1}, …, f_{r}) for certain f_{1}, …, f_{r} ∈ K[X_{1}, …, X_{n}]. Let further f ∈ K[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}] and W = V(f_{1}, …, f_{r}, f) ⊂ K^{n+m}. Suppose that C_{K[X_{1},...,X_{n}]}(f) + I(V) = (1) and assume that there is a Zariski-open set U ⊂ V such that f(a, Y_{1}, …, Y_{m}) is non-constant and irreducible in K̃[Y_{1}, …, Y_{m}] for all a ∈ U. Then

(i) W is an absolutely irreducible K-variety and I(W) = (f_{1}, …, f_{r}, f).

(ii) dim W = dim V + m − 1.

(iii) Let z = (a_{1}, …, a_{n}, b_{1}, …, b_{m}) ∈ W and let a = (a_{1}, …, a_{n}). If z ∈ W_{reg}

then a ∈ V_{reg}. (iv) Let z = (a_{1}, …, a_{n}, b_{1}, …, b_{m}) ∈ W and let a = (a_{1}, …, a_{n}). If a ∈ V_{reg}

and \frac{∂f}{∂Y_{i}} (z) ≠ 0 for some i ∈ \{1, …, m\} then z ∈ W_{reg}.

Proof As V is absolutely irreducible A := K̃[X_{1}, …, X_{n}]/I(V) is a domain. Let F := Quot(A) and denote the residue classes of the indeterminates by x_{1}, …, x_{n}. Then A = K̃[x_{1}, …, x_{n}]. The canonical epimorphism π : K̃[X_{1}, …, X_{n}] ↠ A extends to an epimorphism

π : K̃[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}] ↠ A[Y_{1}, …, Y_{m}]

with π(Y_{i}) = Y_{i} (1 ≤ i ≤ m). This extension induces an isomorphism

π̄ : K̃[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}]/(f_{1}, …, f_{r}, f) ⟶ A[Y_{1}, …, Y_{m}]/(π(f)).

In order to prove (i) we have to show that π(f) generates a prime ideal in A[Y_{1}, …, Y_{m}]. From the assumption C_{K[X_{1},...,X_{n}]}(f) + I(V) = (1) one immediately gets C_{A}(π(f)) = (1) and thus by Proposition A.2 it suffices to prove that π(f) is irreducible in F [Y_{1}, …, Y_{m}].

Assume that π(f) is reducible in F [Y_{1}, …, Y_{m}]. Then there are polynomials g, h ∈ K̃[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}] and p ∈ K̃[X_{1}, …, X_{n}] such that g(x, Y), h(x, Y) are non-constant polynomials in A[Y_{1}, …, Y_{m}], p(x) ≠ 0 and p(x) · f(x, Y) = g(x, Y) · h(x, Y).

Since V is absolutely irreducible and U is Zariski-open there is some a ∈ U such that p(a) ≠ 0 and g(a, Y), h(a, Y) are non-constant polynomials. As a ∈ V there is a K̃-epimorphism φ : A ↠ K̃ such that φ(x_{i}) = a_{i} (1 ≤ i ≤ n). Hence we get p(a)·f(a, Y) = g(a, Y)·h(a, Y). Therefore f(a, Y) is reducible in K̃[Y_{1}, …, Y_{m}] and we got a contradiction. This proves (i).

Obviously W = (V × K̃^{m}) ∩ V(f). Since f(a, Y_{1}, …, Y_{m}) ∈ K̃[Y_{1}, …, Y_{m}] \ K̃ for a ∈ U we get V × K̃^{m} ⊄ V(f) and (V × K̃^{m}) ∩ V(f) ≠ ∅. Hence the dimension theorem yields

dim W = dim(V × K̃^{m}) − 1 = dim V + m − 1.

This proves (ii). In order to prove (iii) and (iv) we have to consider the ranks of the Jacobian matrices J_{V} and J_{W} of the varieties V and W. One can easily see that

J_{W} (z) = ( \begin{matrix}& & & 0 & ⋯ & 0 \\ & J_{V} (a) & & ⋮ & & ⋮ \\ & & & 0 & ⋯ & 0 \\ \frac{∂f}{∂X_{1}} (z) & ⋯ & \frac{∂f}{∂X_{n}} (z) & \frac{∂f}{∂Y_{1}} (z) & ⋯ & \frac{∂f}{∂Y_{m}} (z) \end{matrix} ) .

If z ∈ W_{reg} then rk(J_{W} (z)) = (n + m) − dim W = n − dim V + 1. Thus rk(J_{V} (a)) ≥ rk(J_{W} (z)) − 1 = n − dim V. However, for every x ∈ V the inequality rk(J_{V} (x)) ≤ n − dim V holds. Hence rk(J_{V} (a)) = n − dim V and thus a ∈ V_{reg}.

If a is a regular point of V then rk(J_{V} (a)) = n − dim V. If \frac{∂f}{∂Y_{i}} (z) ≠ 0 for at least one i we get rk(J_{W} (z)) = rk(J_{V} (a)) + 1 = n − dim V + 1 = (n + m) − dim W and hence z ∈ W_{reg}.

References

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