Classification of semi-algebraic p-adic sets up to semi-algebraic bijection.

Raf Cluckers^{∗}

Abstract

We prove that two infinite p-adic semi-algebraic sets are isomorphic (i.e. there exists a semi-algebraic bijection between them) if and only if they have the same dimension.

The real semi-algebraic sets can be classified up to semi-algebraic bijection as follows [4]:

There exists a real semi-algebraic bijection between two real semi-algebraic sets if and only if they have the same dimension and Euler characteristic. More generally L. van den Dries [4] gave such a classification for o-minimal expansions of the real field, using the dimension and Euler characteristic as defined for o-minimal structures. Since this Euler characteristic χ is in fact the canonical map from the real semi-algebraic sets onto the Grothendieck ring of ℝ (which is ℤ), we see that the isomorphism class of a semi-algebraic set only depends on its image in the Grothendieck ring and its dimension.

In this paper we treat the p-adic analogue of this classification. The Grothendieck ring of ℚ_{p} is recently proved to be trivial by D. Haskell and the author [1], so the analogue of the real case is a classification of the p-adic semi-algebraic sets up to semi-algebraic bijection using only the dimension. We give such a classification for the semi-algebraic sets of ℚ_{p} and for finite field extensions of ℚ_{p}, using explicit isomorphisms of [1] and the p-adic Cell Decomposition Theorem of J. Denef [2]. The most difficult part in giving this classification is to prove that for any semi-algebraic set X there is a finite partition into semi-algebraic sets, such that each part is isomorphic to a Cartesian product of one dimensional sets, in other words semi-algebraic sets have a rectilinearization. Since all arguments hold also for finite field extensions of ℚ_{p}, we work in this more general setting.

Let p denote a fixed prime number, ℚ_{p} the field of p-adic numbers and K a fixed finite field extension of ℚ_{p}. For x ∈ K let v(x) ∈ ℤ ∪ {+∞} denote the valuation of x. Let R = {x ∈ K|v(x) ≥ 0} be the valuation ring, K^{×} = K \ {0} and P_{n} = {x ∈ K^{×}|∃y ∈ K y^{n} = x}. We call a subset of K^{n} semi-algebraic if it is a Boolean combination (i.e. obtained by taking finite unions, complements and intersections) of sets of the form {x ∈ K^{m}|f(x) ∈ P_{n}} and {x ∈ K^{m}|f(x) = 0}, with f(x) ∈ K[X_{1}, …, X_{m}]. The collection of semi-algebraic sets is closed under taking projections K^{m} → K^{m−1}, even more: it consists precisely of the projections of affine p-adic varieties (see [2] and [5]). Further we have that sets of the form {x ∈ K^{m}|v(f(x)) ≤ v(g(x))} with f(x), g(x) ∈ K[X_{1}, …, X_{m}] are semi-algebraic (see [2], Lemma 2.1). A function f : A → B is semi-algebraic if its graph is a semi-algebraic set; if further f is a bijection, we call f an isomorphism and we write A ≅ B.

∗Research supported by Fund for Scientific Research, Flanders (Belgium)(F.W.O.).

Let π be a fixed element of R with v(π) = 1, thus π is a uniformizing parameter for R. For a semi-algebraic set X ⊂ K we write

X^{(k)} = {x ∈ X|v(π^{−v(x)}x − 1) ≥ k},

which is semi-algebraic (see [2], Lemma 2.1); X^{(k)} consists of those points x ∈ X which have a p-adic expansion x = ∑_{i=s}^{∞} a_{i}π^{i} with a_{s} = 1 and a_{i} = 0 for i = s + 1, …, k − 1. By a finite partition of a semi-algebraic set we mean a partition into finitely many semi-algebraic sets. Let X ⊂ K^{n}, Y ⊂ K^{m} be semi-algebraic. Choose disjoint semi-algebraic sets X′, Y′ ⊂ K^{k} for some k, such that X ≅ X′ and Y ≅ Y′, then we define the disjoint union of X and Y up to isomorphism as X′ ∪ Y′. In the introduction of [1] it is shown that we can take k = max(m, n), i.e. we can realize the disjoint union without going into higher dimensional affine spaces.

We recall some well-known facts.

Lemma 1 (Hensel) Let f(t) be a polynomial over R in one variable t, and let α ∈ R, e ∈ ℕ. Suppose that f(α) ≡ 0 mod π^{2e+1} and v(f′(α)) ≤ e, where f′ denotes the derivative of f. Then there exists a unique ᾱ ∈ R such that f(ᾱ) = 0 and ᾱ ≡ α mod π^{e+1}.

Corollary 1 Let n > 1 be a natural number. For each k > v(n), and k′ = k + v(n) the function

K^{(k)} → P_{n}^{(k′)} : x ↦ x^{n}

is an isomorphism.

The next theorem gives some concrete isomorphisms between one dimensional sets.

Proposition 1 ([1], Prop. 2)

(i) The union of two disjoint copies of R \ {0} is isomorphic to R \ {0}.

(ii) For each k > 0 the union of two disjoint copies of R^{(k)} is isomorphic to R^{(k)}.

(iii) R ≅ R \ {0}.

We deduce an easy corollary, also consisting of concrete isomorphisms.

Corollary 2 For each k we have isomorphisms

(i) R^{(k)} ≅ R \ {0},

(ii) R \ {0} ≅ K.

Proof. (i) There is a finite partition R \ {0} = ⋃_{α} αR^{(k)} with v(α) = 0, say with s parts. Then R \ {0} is a fortiori isomorphic to the union of s disjoint copies of R^{(k)}, which is by Proposition 1(ii) isomorphic to R^{(k)}. (ii) The map

(\{0\} × R) ∪ (\{1\} × R^{×}) → K : \{ \begin{matrix}(0, x) & ↦ & x, \\ (1, x) & ↦ & 1/(πx), \end{matrix}

is a well-defined isomorphism. It follows that K is isomorphic to the disjoint union of R and R \ {0}. Now use (i) and (iii) of Proposition 1. Q.E.D.

Give K^{m} the topology induced by the norm |x| = max(|x_{i}|_{p}) with |x_{i}|_{p} = p^{−v(x_{i})} for x = (x_{1}, … x_{m}) ∈ K^{m}. P. Scowcroft and L. van den Dries [6] proved there exists no isomorphism from an open set A ⊂ K^{m} onto an open set B ⊂ K^{n} with n ≠ m, so we can define the dimension of semi-algebraic sets as follows.

Definition 1 ([6]) The dimension of a semi-algebraic set X is the greatest natural number n such that we have a semi-algebraic subset A ⊂ X and an isomorphism from A to a semi-algebraic open subset of K^{n}.

P. Scowcroft and L. van den Dries [6] proved many good properties of this dimension, for example that it is invariant under isomorphisms.

Proposition 2 ([6]) Let A and B be semi-algebraic sets, then the following is true:

(i) If A ≅ B then dim(A) = dim(B),

(ii) dim(A ∪ B) = max(dim(A), dim(B)).

We will prove the converse of (i) for infinite semi-algebraic sets.

Lemma 2 For any semi-algebraic set X of dimension m ∈ ℕ_{0} there exists a semi-algebraic injection X → K^{m}.

Proof. By [6], Cor. 3.1 there is a finite partition of X such that each part A is isomorphic to a semi-algebraic open A′ ⊂ K^{k} for some k ≤ m. Now realize the disjoint union of the sets A′ without going into higher embedding dimension (see the introduction). Q.E.D.

We formulate the p-adic Cell Decomposition Theorem by J. Denef [2, 3], which is the analogue of the real semi-algebraic Cell Decomposition Theorem.

Theorem 1 (Cell Decomposition [2, 3]) Let x = (x_{1}, …, x_{m}) and x̂ = (x_{1}, …, x_{m−1}), m > 0. Let f_{i}(x̂, x_{m}), i = 1, …, r, be polynomials in x_{m} with coefficients which are semi-algebraic functions from K^{m−1} to K. Let n ∈ ℕ_{0} be fixed. Then there exists a finite partition of K^{m} into sets A of the form

A = {x ∈ K^{m}|x̂ ∈ D and v(a_{1}(x̂))□_{1}v(x_{m} − c(x̂))□_{2}v(a_{2}(x̂))},

such that

f_{i}(x) = u_{i}(x)^{n}h_{i}(x̂)(x_{m} − c(x̂))^{ν_{i}}, for each x ∈ A, i = 1, …, r,

with u_{i}(x) a unit in R for each x, D ⊂ K^{m−1} semi-algebraic, ν_{i} ∈ ℕ, h_{i}, a_{1}, a_{2}, c semi-algebraic functions from K^{m−1} to K and □_{1}, □_{2} either ≤, <, or no condition.

The next lemma is also due to J. Denef [3].

Lemma 3 ([3], Cor. 6.5) Let b : K^{m} → K be a semi-algebraic function. Then there exists a finite partition of K^{m} such that for each part A we have e > 0 and polynomials f_{1}, f_{2} ∈ R[X_{1}, … X_{m}] such that

v(b(x)) = \frac{1}{e}v(\frac{f_{1}(x)}{f_{2}(x)}), for each x ∈ A,

with f_{2}(x) ≠ 0 for each x ∈ A.

We give an application of the Cell Decomposition Theorem and Lemma 3, inspired by similar applications in [3]. For details of the proof we refer to the proof of [3], Thm. 7.4. By λP_{n} with λ = 0 we mean {0}.

Lemma 4 Let X ⊂ K^{m} be semi-algebraic and b_{j} : K^{m} → K semi-algebraic functions for j = 1, …, r. Then there exists a finite partition of X s.t. each part A has the form

A = {x ∈ K^{m}|x̂ ∈ D, v(a_{1}(x̂))□_{1}v(x_{m} − c(x̂))□_{2}v(a_{2}(x̂)),
x_{m} − c(x̂) ∈ λP_{n}},

and such that for each x ∈ A we have

v(b_{j}(x)) = \frac{1}{e_{j}} v((x_{m} − c(x̂))^{μ_{j}}d_{j}(x̂)),

with x̂ = (x_{1}, …, x_{m−1}), D ⊂ K^{m−1} semi-algebraic, e_{j} > 0, n > 1, μ_{j} ∈ ℤ, λ ∈ K, c, a_{i}, d_{j} semi-algebraic functions from K^{m−1} to K and □_{i} either <, ≤ or no condition.

Proof. By Lemma 3 we have a finite partition of X such that for each part A_{0} we have e_{j} > 0 and polynomials g_{j}, g′_{j} ∈ R[X_{1}, …, X_{m}] with

v(b_{j}(x)) = \frac{1}{e_{j}} v(\frac{g_{j}(x)}{g′_{j}(x)}), for each x ∈ A_{0}, j = 1, …, r.

Let f_{i} be the polynomials which appear in a quantifier free formula which defines A_{0}. Apply now the Cell Decomposition Theorem as in the proof of [3], Thm. 7.4 to the polynomials f_{i}, g_{j} to obtain the lemma. Q.E.D.

The proof of the next proposition is an application of both the Cell Decomposition Theorem and some hidden Presburger arithmetic in the value group of K; it is the technical heart of this paper. If l = 0 then ∏_{i=1}^{l} R^{(k)} denotes the set {0}.

Definition 2 We say that a semi-algebraic function f : B → K satisfies condition (1) (with constants e, μ_{i}, β) if we have constants e ∈ ℕ_{0}, μ_{i} ∈ ℤ, β ∈ K such that each x = (x_{i}) ∈ B satisfies

v(f(x)) = \frac{1}{e}v(β ∏_{i} x_{i}^{μ_{i}}). (1)

Proposition 3 (Rectilinearization) Let X be a semi-algebraic set and b_{j} : X → K semi-algebraic functions for j = 1, …, r. Then there exists a finite partition of X such that for each part A we have constants l ∈ ℕ, k ∈ ℕ_{0}, μ_{ij} ∈ ℤ, β_{j} ∈ K, and an isomorphism

f : ∏_{i=1}^{l} R^{(k)} → A,

such that for each x = (x_{1}, …, x_{l}) ∈ ∏_{i=1}^{l} R^{(k)} we have

v(b_{j} ∘ f(x)) = v(β_{j} ∏_{i=1}^{l} x_{i}^{μ_{ij}}).

Proof. We work by induction on m. Let dim(X) = 1 and b_{j} : X → K semi-algebraic functions, j = 1, …, r. By Lemma 2 we may suppose that X ⊂ K. We reduce first to the case that X and b_{j} have the special form (2) (see below). By Lemma 4 there is a partition such that each part A is either a point or of the form

A = {x ∈ K|v(a_{1})□_{1}v(x − c)□_{2}v(a_{2}), x − c ∈ λP_{n}},

and such that for each x ∈ A we have v(b_{j}(x)) = \frac{1}{e_{j}} v(β_{j}(x − c)^{μ_{j}}), with a_{i}, c, λ, β_{j} ∈ K.

We may assume that λ ≠ 0, a_{1} ≠ 0 ≠ a_{2}, □_{i} is either ≤ or no condition and since the translation

{x ∈ K|v(a_{1})□_{1}v(x)□_{2}v(a_{2}), x ∈ λP_{n}} → A : x ↦ x + c

is an isomorphism, we may also assume that c = 0. If both □_{1} and □_{2} are no condition we can partition A into parts {x ∈ A|0 ≤ v(x)} and {x ∈ A|v(x) ≤ −1}. It follows that if □_{1} is no condition we may suppose that □_{2} is ≤, then we can apply the isomorphism

\{x ∈ K|v(\frac{1}{a_{2}} ) ≤ v(x), x ∈ \frac{1}{λ}P_{n}\} → A : x ↦ \frac{1}{x},

and replace μ_{j} by −μ_{j}.

This shows we can reduce to the case that X has the form

X = {x ∈ K|v(a_{1}) ≤ v(x)□_{2}v(a_{2}), x ∈ λP_{n}}, (2)

with a_{1} ≠ 0 ≠ a_{2}, λ ≠ 0, □_{2} either ≤ or no condition and v(b_{j}(x)) = \frac{1}{e_{j}} v(β_{j}(x)^{μ_{j}}) for each x ∈ X.

Case 1: □_{2} is ≤ (in equation (2)). By Hensel’s Lemma we can partition X into finitely many parts of the form y + π^{s}R for some fixed s > v(a_{2}) and with v(a_{1}) ≤ v(y) ≤ v(a_{2}) for each y. For each such part there is a finite partition y + π^{s}R = ⋃_{γ∈Γ} A_{γ} ∪ {y}, with A_{γ} = y + π^{s}γR^{(1)} and v(γ) = 0 for each γ. The functions f_{γ} : R^{(1)} → A_{γ} : x ↦ y + π^{s}γx are isomorphisms which satisfy v(b_{j} ∘ f_{γ}(x)) = \frac{1}{e_{j}} v(β_{j}y^{μ_{j}}) for all x ∈ R^{(1)}. This last expression is independent of x, so there exists β′_{j} ∈ K such that v(b_{j} ∘ f_{γ}(x)) = v(β′_{j}) for all x ∈ R^{(1)}. This proves Case 1.

Case 2: □_{2} is no condition (in equation (2)). The map

f_{1} : R ∩ λ′P_{n} → X : x ↦ a_{1}x,

with λ′ = λ/a_{1} is an isomorphism. Let n′ be a common multiple of e_{1}, …, e_{r} and n. Choose k > v(n′) and put k′ = k + v(n′). Let R ∩ λ′P_{n} = ⋃_{γ} B_{γ} be

a finite partition, with B_{γ} = γ(R ∩ P_{n′}^{(k′)}) and 0 ≤ v(γ) < n′. Now we have that the map f_{γ} : R^{(k)} → B_{γ} : x ↦ γx^{n′} is an isomorphism by corollary 1.

Let g_{γ} be the semi-algebraic function f_{1} ∘ f_{γ}, which is an isomorphism from R^{(k)} onto a semi-algebraic set A_{γ} ⊂ X. The sets A_{γ} form a finite partition of X. Put μ′_{j} = μ_{j}n′/e_{j}, then we have for each x ∈ R^{(k)} that

v(b_{j} ∘ g_{γ}(x)) = \frac{1}{e_{j}} v(β_{j}(a_{1}γx^{n′})^{μ_{j}}) = \frac{1}{e_{j}} v(β_{j}(a_{1}γ)^{μ_{j}}) + v(x^{μ′_{j}})

= v(β′_{j}) + v(x^{μ′_{j}}) = v(β′_{j}x^{μ′_{j}}),

with β′_{j} in K such that v(β′_{j}) = \frac{1}{e_{j}} v(β_{j}(a_{1}γ)^{μ_{j}}). This proves case 2.

Now let dim(X) = m > 1 and let b_{j} : X → K be semi-algebraic functions, j = 1, …, r. By Lemma 2 we may suppose that X ⊂ K^{m}. Claim. We can partition X such that for each part A we have an isomorphism of the form f : D_{1} × D_{m−1} → A, with D_{1} ⊂ K and D_{m−1} ⊂ K^{m−1} semi-algebraic, such that the functions b_{j} ∘ f satisfy condition (1), i.e. there are constants e_{j} ∈ ℕ_{0}, μ_{ij} ∈ ℤ, β_{j} ∈ K such that each x = (x_{i}) ∈ D_{1} × D_{m−1} satisfies

v(b_{j} ∘ f(x)) = \frac{1}{e_{j}} v(β_{j} ∏_{i} x_{i}^{μ_{ij}}).

If the claim is true, we can apply the induction hypotheses once to D_{1} and the functions x_{1} ↦ x_{1}^{μ_{1j}} and once to D_{m−1} and the functions (x_{2}, …, x_{m}) ↦ β_{j} ∏_{i=2}^{m} x_{i}^{μ_{ij}} for j = 1, …, r. It follows easily that we can partition X such that for each part A there is an isomorphism f : ∏_{i} R^{(k)} → A such that all f ∘ b_{j} satisfy condition (1) with constants e′_{j}, μ′_{ij} and β′_{j}. Now we can proceed as in Case 2 for m = 1 to make all e′_{j} ∈ ℕ_{0} occurring in condition (1) equal to 1. The proposition follows now immediately.

Proof of the claim. First we show we can reduce to the case described in equation (3) below. Using Lemma 4 we find a finite partition of X such that each part A has the form

A = {x ∈ K^{m}|x̂ ∈ D, v(a_{1}(x̂))□_{1}v(x_{m} − c(x̂))□_{2}v(a_{2}(x̂)),
x_{m} − c(x̂) ∈ λP_{n}},

and such that for each x ∈ A we have v(b_{j}(x)) = \frac{1}{e_{j}} v((x_{m} − c(x̂))^{μ_{mj}}d_{j}(x̂)), with μ_{mj} ∈ ℤ. Similar as for m = 1, we may suppose that c(x̂) = 0 for all x̂. Apply now the induction hypotheses to the set D ⊂ K^{m−1} and the functions a_{1}, a_{2}, d_{j}. We find a finite partition of A such that for each part A′ we have an isomorphism f : B → A′, where B is a set of the form

B = {x ∈ K^{m}|x̂ ∈ D′, v(α_{1} ∏_{i=1}^{l} x_{i}^{η_{i}})□_{1}v(x_{m})□_{2}v(α_{2} ∏_{i=1}^{l} x_{i}^{ε_{i}}), x_{m} ∈ λP_{n}},

with D′ = ∏_{i=1}^{l} R^{(k)}, l ≤ m − 1, such that each b_{j} ∘ f satisfies condition (1). We will alternately partition further and apply isomorphisms to the parts which compositions with b_{j} will always satisfy condition (1). By the induction hypotheses we may suppose that λ ≠ 0 and dim(D′) = m − 1, i.e. D′ = ∏_{i=1}^{m−1} R^{(k)}. Analogously as for m = 1 we may suppose that α_{1} ≠ 0 ≠ α_{2}, □_{2} is either ≤ or no condition and □_{1} is the symbol ≤ (possibly after partitioning or applying x ↦ (x_{1}, …, x_{m−1}, 1/x_{m})).

Choose k̄ > v(n) and put k′ = k̄ + v(n). We may suppose that k′ > k, so we have a finite partition B = ⋃_{γ} B_{γ} with γ = (γ_{1}, …, γ_{m}) ∈ K^{m},

0 ≤ v(γ_{i}) < n and B_{γ} = {x ∈ B|x_{i} ∈ γ_{i}P_{n}^{(k′)}}. Now we have isomorphisms

f_{γ} : C_{γ} → B_{γ} : x ↦ (γ_{1}x_{1}^{n}, …, γ_{m}x_{m}^{n}),

with

C_{γ} = {x ∈ ( ∏_{i=1}^{m−1} R^{(k̄)}) × K^{(k̄)}|v(α′_{1} ∏_{i=1}^{m−1} x_{i}^{η_{i}}) ≤ v(x_{m})□_{2}v(α′_{2} ∏_{i=1}^{m−1} x_{i}^{ε_{i}})},

for appropriate choice of α′_{i} ∈ K. Put ν_{i} = ε_{i} − η_{i}, β = α′_{2}/α′_{1}, then we have the isomorphism

\begin{matrix}\{x ∈ ∏_{i=1}^{m} R^{(k̄)}|v(x_{m})□_{2}v(β ∏_{i=1}^{m−1} x_{i}^{ν_{i}})\} → C_{γ} : \\ x ↦ (x_{1}, …, x_{m−1}, α′_{1}x_{m} ∏_{i=1}^{m−1} x_{i}^{η_{i}}). \end{matrix}

If □_{2} is no condition, the claim is trivial. It follows that we can reduce to the case that we have an isomorphism

f : E = {x ∈ ∏_{i=1}^{m} R^{(k̄)}|v(x_{m}) ≤ v(β ∏_{i=1}^{m−1} x_{i}^{ν_{i}})} → X (3)

with β ≠ 0, such that each b_{j} ∘ f satisfies condition (1).

Suppose we are in the case described in (3). If ν_{i} ≤ 0 for i = 1, …, m−1 then we have a finite partition E = ∪_{s}E^{=s}, with s ∈ {0, 1, …, v(β)} and E^{=s} = {x ∈ E|v(x_{m}) = s}. Also, E^{=s} = {(x_{1}, …, x_{m−1})|∃x_{m}, (x_{1}, …, x_{m}) ∈ E^{=s}} × {x_{m} ∈ R^{(k̄)}|v(x_{m}) = s} and the claim follows.

Suppose now that ν_{1} > 0 in (3). First we prove the proposition when ν_{1} = 1, using some implicit Presburger arithmetic on the value group. We can partition E into parts E_{1} and E_{2}, with

E_{1} = {x ∈ E|v(x_{m}) < v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}})},

E_{2} = {x ∈ E|v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}) ≤ v(x_{m})}

= {x ∈ ∏_{i=1}^{m} R^{(k̄)}|v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}) ≤ v(x_{m}) ≤ v(βx_{1} ∏_{i=2}^{m−1} x_{i}^{ν_{i}})}.

Since v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}) ≤ v(x_{1}β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}) for x ∈ E_{1}, it follows that

E_{1} = R^{(k̄)} × {(x_{2}, …, x_{m}) ∈ ∏_{i=2}^{m} R^{(k̄)}|v(x_{m}) < v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}})},

and the restrictions b_{j} ∘ f|E_{1} satisfy condition (1). As for E_{2}, let D_{m−1} be the set

D_{m−1} = {(x_{2}, …, x_{m}) ∈ ∏_{i=2}^{m} R^{(k̄)}|v(β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}) ≤ v(x_{m})}.

We may suppose that β ∈ K^{(k̄)}, then the map

R^{(k̄)} × D_{m−1} → E_{2} : x ↦ (\frac{x_{1}x_{m}}{β ∏_{i=2}^{m−1} x_{i}^{ν_{i}}} , x_{2}, …, x_{m}),

can be checked by elementary Presburger arithmetic to be an isomorphism. This proves the claim when ν_{1} = 1.

Suppose now that X is of the form described in (3) and ν_{1} > 1. We prove we can reduce to the case ν_{1} = 1 by partitioning and applying appropriate power maps. Choose k̃ > v(ν_{1}) and put k̃′ = k̃+v(ν_{1}). We may suppose that k̃ ≥ k̄, so we have a finite partition E = ⋃_{α} E_{α}, with α = (α_{1}, …, α_{m}) ∈ K^{m}, v(α_{1}) = 0, 0 ≤ v(α_{i}) < ν_{1} for i = 2, …, m and

E_{α} = {x ∈ E|x_{1} ∈ α_{1}R^{(k̃)}, x_{i} ∈ α_{i}P_{ν_{1}}^{(k̃′)} for i = 2, …, m}.

By corollary 1 we have isomorphisms

f_{α} : C_{α} → E_{α} : x ↦ (α_{1}x_{1}, α_{2}x_{2}^{ν_{1}}, …, α_{m}x_{m}^{ν_{1}}),

with C_{α} = {x ∈ ∏_{i=1}^{m} R^{(k̃)}|v(x_{m}) ≤ v(β′x_{1} ∏_{i=2}^{m−1} x_{i}^{ν_{i}})}, where β′ ∈ K^{×} depends on α. This reduces the problem to the case described in (3) with ν_{1} = 1 and thus the proposition is proved. Q.E.D.

Remark 1 Proposition 3 can be strengthened, adding conditions on the Jacobians of the isomorphisms, to become useful to p-adic integration theory.

Theorem 2 Let X be a semi-algebraic set, then either X is finite or there exists a semi-algebraic bijection X → K^{k} with k ∈ ℕ_{0} the dimension of X.

Proof. We give a proof by induction on dim(X) = m. Let dim(X) = 1. Use Proposition 3 to partition X such that each part is isomorphic to either R^{(k)} or a point. By combining the isomorphisms of Proposition 1 and Corollary 2, it follows that X ≅ K.

Now suppose dim(X) = m > 1. Proposition 3 together with the case m = 1 implies that we can finitely partition X such that each part is isomorphic to K^{l}, for some l ∈ {0, …, m}, with K^{0} = {0}. Suppose that A and B are disjoint parts, such that A ≅ K^{l} and B ≅ K^{k}, with 1 ≤ k ≤ m and l ≤ k. It is enough to prove that A ∪ B ≅ K^{k}. We prove this by induction on l. Suppose l = 0, so A = {a}. Since B ≅ K^{k} and k ≥ 1 there exists an injective semi-algebraic function i : R → A∪B such that i(R\{0}) ⊂ B and i(0) = a. It follows that A∪B ≅ B ≅ K^{k} since R ≅ R \{0} (Proposition 1). If l = k = 1 then dim(A ∪ B) = 1 thus we can use the case m = 1 to find A ∪ B ≅ K. If 1 ≤ l and 1 < k we have A ∪ B ≅ K × (A′ ∪ B′), for some disjoint sets A′ ≅ K^{l−1} and B′ ≅ K^{k−1}. Use induction on l to find A′∪B′ ≅ K^{k−1} and thus A∪B ≅ K^{k}. This proves Theorem 2. Q.E.D.

We obtain as a corollary of Theorem 2 the following classification of the p-adic semi-algebraic sets.

Corollary 3 Two infinite semi-algebraic sets are isomorphic if and only if they have the same dimension.

Remark 2 The field 𝔽_{q}((t)) of Laurent series over the finite field is often considered to be the characteristic p counterpart of the p-adic numbers. D. Haskell and the author [1] proved that the Grothendieck ring of 𝔽_{q}((t)) in the language of rings is trivial, analogously as for Q_{p}. In [1], proof of Thm. 2, it is also shown that there is a definable injection from the plane 𝔽_{q}((t))^{2} into the line 𝔽_{q}((t)), which makes it plausible we cannot define an invariant for this field which behaves like a (good) dimension. It is an open question to classify the definable sets of 𝔽_{q}((t)) up to definable bijection.

References

[1] Cluckers, R.; Haskell, D.: Grothendieck rings of ℤ-valued fields, Bull. Symbolic Logic (to appear).

[2] Denef, J.: p-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1986), 154-166.

[3] Denef, J.: The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984), 1-23.

[4] van den Dries, L.: Tame topology and o-minimal structures, LMS Lecture Note Series 248 CUP (1998).

[5] Macintyre, A.: On definable subsets of p-adic fields, J. Symb. Logic 41 (1976), 605-610.

[6] Scowcroft, P.; van den Dries, L.: On the structure of semialgebraic sets over p-adic fields, J. Symbolic Logic 53 (1988), no. 4, 1138-1164.

Raf Cluckers, Katholieke Universiteit Leuven Celestijnenlaan 200B B-3001 Heverlee, Belgium; e-mail address: raf.cluckers@wis.kuleuven.ac.be