Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

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Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

Jean-Baptiste Campesato

^∗

Toshizumi Fukui

^†

Krzysztof Kurdyka

^{‡ ¶}

Adam Parusiński

^{§ ¶}

December 14, 2018

Abstract

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps.

As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs.

Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

2010 Mathematics Subject Classification. Primary: 14P99. Secondary: 26A16, 14E18, 14B05.

Keywords: Real algebraic geometry, motivic integration, singularities, Lipschitz maps, Nash functions, arc-analytic maps, arc-symmetric sets, virtual Poincaré polynomial.

∗Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France E-mail:Jean-Baptiste.Campesato@univ-amu.fr

†Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570, Japan

E-mail:tfukui@rimath.saitama-u.ac.jp

‡Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France E-mail:Krzysztof.Kurdyka@univ-savoie.fr

§Université Côte d’Azur, Université Nice Sophia Antipolis, CNRS, LJAD, France E-mail:Adam.Parusinski@unice.fr

¶Partially supported by ANR project LISA (ANR-17-CE40-0023-03).

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1 Introduction

In this paper we establish relations between the arc space and the Lipschitz geometry of a singular real algebraic variety.

The interest in the Lipschitz geometry of real analytic and algebraic spaces emerged in the 70’s of the last century by a conjecture of Siebenmann and Sullivan: there are only countably many local Lipschitz structures on real analytic spaces. Subsequently the Lip- schitz geometry of real and complex algebraic singularities attracted much attention and various methods have been developed to study it: stratification theory [33,37],𝐿-regular decompositions [23,38,25,42], Lipschitz triangulations [46], non-archimedean geometry [16], and recently, in the complex case, resolution and low dimensional topology [4]. In the algebraic case Siebenmann and Sullivan’s conjecture was proved in [36]. The general analytic case was solved in [47].

In this paper we study various versions of Lipschitz inverse mapping theorems, with respect to the inner distance, for homeomorphisms𝑓 ∶𝑋→𝑌 between (possibly singular) real algebraic set germs. Recall that a connected real algebraic, and more generally a connected semialgebraic, subset𝑋 ⊂ℝ^𝑁is path-connected (by rectifiable curves), so we have aninnerdistance on𝑋, defined by the infimum over the length of rectifiable curves joining two given points in𝑋.

We assume that the homeomorphism𝑓 is semialgebraic and generically arc-analytic.

For instance the recently studied continuous rational maps [21,20,19] are of this type.

Arc-analytic mappings were introduced to real algebraic geometry in [22]. Those are the mappings sending by composition real analytic arcs to real analytic arcs. It was shown in

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[2,39] that the semialgebraic arc-analytic mappings coincide with the blow-Nash mappings.

Moreover, by [41], real algebraic sets admit algebraic stratifications with local semialgebraic arc-analytic triviality along each stratum.

What we prove can be stated informally as follows: if𝑓⁻¹is Lipschitz, then so is𝑓 itself.

The problem is non-trivial even when the germs(𝑋, 𝑥)and (𝑌 , 𝑦) are non-singular [14].

When these germs are singular, then the problem is much more delicate. In fact we have to assume that the motivic measures of the real analytic arcs drawn on(𝑋, 𝑥)and(𝑌 , 𝑦)are equal.

Developing a rigorous theory of motivic measure on the space of real analytic arcs for real algebraic sets is another main goal of this paper.

We state below a concise version of our main results. For more precise and more general statements see Theorems4.13and5.10.

Theorem. Let𝑓 ∶ (𝑋, 𝑥) → (𝑌 , 𝑦)be the germ of a semialgebraic generically arc-analytic home- omorphism between two real algebraic set germs, that are of pure dimension¹𝑑. Assume that the motivic measures of the real analytic arcs centered at𝑥in𝑋and of the real analytic arcs centered at 𝑦in𝑌 are equal (see Section3for the definition of the motivic measure). Then

1. If the Jacobian determinant of𝑓is bounded from below then it is bounded from above and𝑓⁻¹ is generically arc-analytic.

2. If the inverse𝑓⁻¹of𝑓 is Lipschitz with respect to the inner distance then so is𝑓.

The proof of this theorem is based on motivic integration. Recall that in the case of complex algebraic varieties, motivic integration was introduced by M. Kontsevitch for non- singular varieties in order to avoid the use of𝑝-adic integrals. Then the theory was developped and extended to the singular case in [10,1,11,28]. The motivic measure is defined on the space of formal arcs drawn on an algebraic variety and takes values in a Grothendieck ring which encodes all the additive invariants of the underlying category. One main ingredi- ent consists in reducing the study to truncated arcs in order to work with finite dimensional spaces. Notice that since the seminal paper of Nash [35], it has been established that the arc space of a variety encodes a great deal of information about its singularities.

In the real algebraic set-up, arguments coming from motivic integration were used in [18,12,8,9] to study and classify the singularities of real algebraic function germs.

In the present paper we construct a motivic measure and a motivic integral for possibly singular real algebraic varieties. Similarly to the complex case, the motivic integral comes together with a change of variables formula which is convenient to do actual computations in terms of resolution of singularities. In our real algebraic set-up this formula holds for generically one-to-one Nash maps and not merely for the birational ones.

A first difference of the present construction compared to the complex one, is that we work with real analytic arcs and not with all formal arcs. However, thanks to Artin approximation theorem, this difference is minor. More importantly, it is not possible to follow exactly the construction of the motivic measure in the complex case because of several ad- ditional difficulties arising from the absence in the real set-up of the Nullstellensatz and of the theorem of Chevalley (the image of a Zariski-constructible set by a regular mapping is Zariski-constructible).

1For ease of reading, in the introduction we avoid varieties admitting points which have a structure of smooth submanifold of smaller dimension as in the handle of the Whitney umbrella{𝑥²=𝑧𝑦²}⊂ℝ³.

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The real motivic measure and the real motivic integral are constructed and studied in Section3.

2 Geometric framework

Throughout this paper, we say that a subset𝑋 ⊂ℝ^𝑁is an algebraic set if it is closed for the Zariski topology, i.e.𝑋may be described as the intersection of the zero sets of polynomials with real coefficients. We denote by𝐼(𝑋)the ideal ofℝ[𝑥₁,…, 𝑥_𝑁]consisting of the polynomials vanishing on𝑋. By noetherianity, we may always assume that the above intersection is indexed by a finite set²and that𝐼(𝑋) = (𝑓₁,…, 𝑓_𝑠)is finitely generated. The dimension dim𝑋of𝑋is the dimension of the ringP(𝑋) = ℝ[𝑥₁,…, 𝑥_𝑁]∕𝐼(𝑋)of polynomial functions on𝑋.

The ringR(𝑋)of regular functions on𝑋is given by the localization ofP(𝑋)with respect to the multiplicative set{ℎ∈P(𝑋), ℎ⁻¹(0) = ∅}. Regular maps are the morphisms of real algebraic sets.

Unless otherwise stated, we will always use the Euclidean topology and not the Zariski one (for instance for the notions of homeomorphism, map germ or closure).

We say that a𝑑-dimensional algebraic set 𝑋 is non-singular at 𝑥 ∈ 𝑋 if there exist 𝑔₁,…, 𝑔_𝑁−𝑑 ∈𝐼(𝑋)and an Euclidean open neighborhood𝑈of𝑥inℝ^𝑁 such that𝑈∩𝑋 = 𝑈∩𝑉(𝑔₁,…, 𝑔_𝑁−𝑑)andrank

(

𝜕𝑔_𝑖

𝜕𝑥_𝑗(𝑥) )

= 𝑁−𝑑. Then there exists an open semialgebraic neighborhood of𝑥in𝑉 which is a𝑑-dimensional Nash submanifold. Notice that the con- verse doesn’t hold [6, Example 3.3.12.b.]. We denote by Reg(𝑋) the set of non-singular points of𝑋. We denote by 𝑋_sing = 𝑋 ⧵Reg(𝑋) the set of singular points of𝑋, it is an algebraic subset of strictly smaller dimension, see [6, Proposition 3.3.14].

A semialgebraic subset ofℝ^𝑁is the projection of an algebraic subset ofℝ^𝑁+𝑚for some 𝑚∈ℕ_≥₀. Actually, by a result of Motzkin [34], we may always assume that𝑚= 1. Equiva- lently, a subset𝑆 ⊂ℝ^𝑁is semialgebraic if and only if there exist polynomials𝑓_𝑖, 𝑔_𝑖,1,…, 𝑔_𝑖,𝑠

𝑖 ∈ ℝ[𝑥₁,…, 𝑥_𝑁]such that

𝑆=

⋃𝑟

𝑖=1

{

𝑥∈ℝ^𝑁, 𝑓_𝑖(𝑥) = 0, 𝑔_𝑖,1(𝑥)>0,…, 𝑔_𝑖,𝑠

𝑖(𝑥)>0} .

Notice that semialgebraic sets are closed under union, intersection and cartesian product.

They are also closed under projection by the Tarski–Seidenberg Theorem. A function is semialgebraic if so is its graph.

We refer the reader to [6] for more details on real algebraic geometry.

Let𝑋 be a non-singular real algebraic set and𝑓 ∶ 𝑋 → ℝ. We say that𝑓 is a Nash function if it is𝐶^∞and semialgebraic. Since a semialgebraic function satisfies a non-trivial polynomial equation and since a smooth function satisfying a non-trivial real analytic equation is real analytic [29,45,5], we obtain that𝑓 is Nash if and only if𝑓 is real analytic and satisfies a non-trivial polynomial equation.

A subset of a real analytic variety is said to be arc-symmetric in the sense of [22] if, given a real analytic arc, either the arc is entirely included in the set or it meets the set at isolated

2Actually, noticing that𝑓₁=⋯=𝑓_𝑠= 0⇔𝑓²

1 +⋯+𝑓_𝑠²= 0, we may always describe a real algebraic set as the zero-set of only one polynomial.

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points only. We are going to work with a slightly different notion defined in [40]. We define AS^𝑁as the boolean algebra generated by semialgebraic³arc-symmetric subsets ofℙ^𝑁

ℝ. We set

AS= ⋃

𝑁∈ℕ_≥0

AS^𝑁. Formally, a subset𝐴 ⊂ℙ^𝑁

ℝ is anAS-set if it is semialgebraic and if, given a real analytic arc𝛾 ∶ (−1,1)→ℙ^𝑁

ℝ such that𝛾(−1,0)⊂ 𝐴, there exists𝜀 >0such that𝛾(0, 𝜀)⊂ 𝐴. Notice that closedAS-subsets ofℙ^𝑁

ℝ are exactly the closed sets of a noetherian topology.

For more on arc-symmetric andASsets we refer the reader to [26].

One important property of theAS sets that we rely on throughout this paper is that it admits an additive invariant richer than the Euler characteristic with compact support, namely the virtual Poincaré polynomial presented later in Section3.2. This is in contrast to the semialgebraic sets, for which, by a theorem of R. Quarez [44], every additive homeomorphism invariant of semialgebraic sets factorises through the Euler characteristic with compact support.

Let𝐸, 𝐵, 𝐹 be threeAS-sets. We say that𝑝∶𝐸→𝐵is anASpiecewise trivial fibration with fiber𝐹 if there exists a finite partition𝐵=⊔𝐵_𝑖intoAS-sets such that𝑝⁻¹(𝐵_𝑖) ≃𝐵_𝑖×𝐹 where≃means bijection withAS-graph.

Notice that, thanks to the noetherianity of theAS-topology, if𝑝∶𝐸→𝐵is locally trivial with fiber𝐹 for theAS-topology⁴, then it is anASpiecewise trivial fibration.

3 Real motivic integration

This section is devoted to the construction of a real motivic measure. Notice that a first step in this direction was done by R. Quarez in [44] using the Euler characteristic with compact support for semialgebraic sets. The measure constructed in this section takes advantage of theAS-machinery in order to use the virtual Poincaré polynomial which is a real analogue of the Hodge–Deligne polynomial in real algebraic geometry. This additive invariant is richer than the Euler characteristic since it encodes, for example, the dimension.

Since real algebraic geometry is quite different from complex algebraic geometry as there is, for example, no Nullstellensatz or Chevalley’s theorem, the classical construction of the motivic measure does not work as it is in this real context and it is necessary to carefully handle these differences.

3.1 Real arcs and jets

We follow the notations of [8, §2.4].

Definition 3.1. The space of real analytic arcs onℝ^𝑁is defined as L(ℝ^𝑁) ={

𝛾∶ (ℝ,0)→ℝ^𝑁, 𝛾real analytic}

3A subset ofℙ^𝑁_ℝis semialgebraic if it is forℙ^𝑁_ℝ seen as an algebraic subset of someℝ^𝑀, or, equivalently, if the intersection of the set with each canonical affine chart is semialgebraic.

4i.e. for every𝑥∈𝐵there is𝑈 ⊂ 𝐵anAS-open subset containing𝑥such that𝑝⁻¹(𝑈) ≃𝑈×𝐹.

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Definition 3.2. For𝑛∈ℕ_≥₀, the space of𝑛-jets onℝ^𝑁is defined as L_𝑛(ℝ^𝑁) = L(ℝ^𝑁)/

∼_𝑛 where𝛾₁∼_𝑛𝛾₂⇔𝛾₁≡𝛾₂ mod𝑡^𝑛+1.

Notation 3.3. For𝑚 > 𝑛, we consider the followingtruncation maps:

𝜋_𝑛∶L(ℝ^𝑁)→L_𝑛(ℝ^𝑁) and

𝜋^𝑚_𝑛 ∶L_𝑚(ℝ^𝑁)→L_𝑛(ℝ^𝑁).

Definition 3.4. For an algebraic set𝑋 ⊂ℝ^𝑁, we define the space of real analytic arcs on𝑋 as

L(𝑋) ={

𝛾∈L(ℝ^𝑁),∀𝑓 ∈𝐼(𝑋), 𝑓(𝛾(𝑡)) = 0} and the space of𝑛-jets on𝑋as

L_𝑛(𝑋) ={

𝛾 ∈L_𝑛(ℝ^𝑁),∀𝑓 ∈𝐼(𝑋), 𝑓(𝛾(𝑡))≡0 mod𝑡^𝑛+1} . The truncation maps induce the maps

𝜋_𝑛∶L(𝑋)→L_𝑛(𝑋) and

𝜋_𝑛^𝑚∶L_𝑚(𝑋)→L_𝑛(𝑋).

Remark 3.5.Notice thatL_𝑛(𝑋)is a real algebraic variety. Indeed, let𝑓 ∈𝐼(𝑋)and𝑎₀,…, 𝑎_𝑛∈ ℝ^𝑁, then we have the following expansion

𝑓(𝑎₀+𝑎₁𝑡+⋯+𝑎_𝑛𝑡^𝑛) =𝑃₀^𝑓(𝑎₀,…, 𝑎_𝑛) +𝑃₁^𝑓(𝑎₀,…, 𝑎_𝑛)𝑡+⋯+𝑃_𝑛^𝑓(𝑎₀,…, 𝑎_𝑛)𝑡^𝑛+⋯ where the coefficients𝑃_𝑖^𝑓 are polynomials. HenceL_𝑛(𝑋)is the algebraic subset ofℝ^𝑁(𝑛+1) defined as the zero-set of the polynomials𝑃_𝑖^𝑓 for𝑓 ∈𝐼(𝑋)and𝑖∈ {0,…, 𝑛}.

In the same way, we may think ofL(𝑋)as an infinite-dimensional algebraic variety.

Remark 3.6. When𝑋is non-singular the following equality holds:

L_𝑛(𝑋) =𝜋_𝑛(L(𝑋))

Indeed, using Hensel’s lemma, we may always lift an𝑛-jet to a formal arc on𝑋and then use Artin approximation theorem to find an analytic arc whose expansion coincides up to the degree𝑛+1. However this equality doesn’t hold anymore when𝑋is singular as highlighted in [8, Example 2.30]. Hence it is necessary to distinguish the spaceL_𝑛(𝑋)of𝑛-jets on𝑋and the space𝜋_𝑛(L(𝑋))⊂L_𝑛(𝑋)of𝑛-jets on𝑋which may be lifted to real analytic arcs on𝑋. We have the following exact statement.

Proposition 3.7([8, Proposition 2.31]). Let𝑋be an algebraic subset ofℝ^𝑁. Then the following are equivalent :

(i) 𝑋is non-singular.

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(ii) ∀𝑛∈ℕ_≥₀,𝜋_𝑛∶L(𝑋)→L_𝑛(𝑋)is surjective.

(iii) ∀𝑛∈ℕ_≥₀,𝜋_𝑛^𝑛+1 ∶L_𝑛+1(𝑋)→L_𝑛(𝑋)is surjective.

Proposition 3.8([8, Proposition 2.33]). Let𝑋be a𝑑-dimensional algebraic subset ofℝ^𝑁. Then (1) For𝑚≥𝑛, the dimensions of the fibers of𝜋_𝑛^𝑚_|_𝜋

𝑚(L(𝑋)) ∶𝜋_𝑚(L(𝑋))→𝜋_𝑛(L(𝑋))are smaller than or equal to(𝑚−𝑛)𝑑.

(2) The fiber( 𝜋_𝑛^𝑛+1)−1

(𝛾)of𝜋_𝑛^𝑛+1 ∶L_𝑛+1(𝑋)→L_𝑛(𝑋)is either empty or isomorphic to𝑇^Zar

𝛾(0)𝑋. Theorem 3.9(A motivic corollary of Greenberg Theorem). Let𝑋 ⊂ℝ^𝑁an algebraic subset.

There exists𝑐∈ℕ_>0(depending only on𝐼(𝑋)) such that

∀𝑛∈ℕ_≥₀, 𝜋_𝑛(L(𝑋)) =𝜋_𝑛^𝑐𝑛(L_𝑐𝑛(𝑋)) Proof. Assume that𝐼(𝑋) = (𝑓₁,…, 𝑓_𝑠).

By the main theorem of [15], there exist𝑁∈ℕ_>0,𝑙∈ℕ_>0and𝜎∈ℕ_≥0(depending only on the ideal ofℝ{𝑡}[𝑥₁,…, 𝑥_𝑁]generated by 𝑓_𝑖 ∈ ℝ[𝑥₁,…, 𝑥_𝑁] ⊂ ℝ{𝑡}[𝑥₁,…, 𝑥_𝑁]) such that∀𝜈≥𝑁 ,∀𝛾 ∈ℝ{𝑡}^𝑁, if𝑓₁(𝛾(𝑡))≡⋯≡𝑓_𝑠(𝛾(𝑡))≡0 mod𝑡^𝜈, then there exists𝜂∈ℝ{𝑡}^𝑁 such that𝜂(𝑡)≡𝛾(𝑡) mod𝑡

⌊𝜈 𝑙

⌋

−𝜎and𝑓₁(𝜂(𝑡)) =⋯=𝑓_𝑠(𝜂(𝑡)) = 0.

Fix𝑐= max (𝑙(𝜎+ 2), 𝑁). We are going to prove that

∀𝑛∈ℕ_≥₀, 𝜋_𝑛(L(𝑋)) =𝜋_𝑛^𝑐𝑛(L_𝑐𝑛(𝑋)) It is enough to prove that𝜋^𝑐𝑛_𝑛 (L_𝑐𝑛(𝑋))⊂ 𝜋_𝑛(L(𝑋))for𝑛≥1.

Let𝑛 ≥1. Let ̃𝛾 ∈ L_𝑐𝑛(𝑋). Then there exists𝛾 ∈ ℝ{𝑡}^𝑁 such that𝛾(𝑡) ≡ ̃𝛾(𝑡) mod𝑡^𝑐𝑛+1 and

𝑓₁(𝛾(𝑡))≡⋯≡𝑓_𝑠(𝛾(𝑡))≡0 mod𝑡^𝑐𝑛+1

Notice that𝑐𝑛+ 1≥𝑁so that there exists𝜂∈ℝ{𝑡}^𝑁such that𝜂(𝑡)≡𝛾(𝑡) mod𝑡

⌊_𝑐𝑛+1

𝑙

⌋

−𝜎

and𝑓₁(𝜂(𝑡)) =⋯=𝑓_𝑠(𝜂(𝑡)) = 0.

Since ⌊𝑐𝑛+ 1

𝑙

⌋

−𝜎 > 𝑛

we have that𝜋_𝑛^𝑐𝑛(̃𝛾) =𝜋_𝑛(𝜂) ∈𝜋_𝑛(L(𝑋)). ■

Remark 3.10. By Tarski–Seidenberg theorem,𝜋_𝑛(L(𝑋)) = 𝜋_𝑛^𝑐𝑛(L_𝑐𝑛(𝑋))is semialgebraic as the projection of an algebraic set. However,𝜋_𝑛(L(𝑋))may not beAS(and thus not Zariski- constructible) as shown in [8, Example 2.32].

This is a major difference with the complex case where𝜋_𝑛(L(𝑋))is Zariski-constructible by Chevalley theorem as the projection of a complex algebraic variety.

Definition 3.11. Let𝑋be an algebraic subset ofℝ^𝑁. We define the ideal𝐻_𝑋ofℝ[𝑥₁,…, 𝑥_𝑁] by

𝐻_𝑋 = ∑

𝑓₁,…,𝑓_𝑁−𝑑∈𝐼(𝑋)

Δ(𝑓₁,…, 𝑓_𝑁−𝑑)((𝑓₁,…, 𝑓_𝑁−𝑑) ∶𝐼(𝑋)) where

• 𝑑= dim𝑋

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• Δ(𝑓₁,…, 𝑓_𝑁−𝑑)is the ideal generated by the𝑁−𝑑minors of the Jacobian matrix (𝜕𝑓_𝑖

𝜕𝑥_𝑗 )

𝑖=1,…,𝑁−𝑑 𝑗=1,…,𝑁

• ((𝑓₁,…, 𝑓_𝑁−𝑑) ∶𝐼(𝑋)) ={

𝑔∈ℝ[𝑥₁,…, 𝑥_𝑁], 𝑔𝐼(𝑋)⊂(𝑓₁,…, 𝑓_𝑁−𝑑)}

is the ideal quotient of the ideal(𝑓₁,…, 𝑓_𝑁−𝑑)by the ideal𝐼(𝑋)

Remark 3.12. By [8, Lemma 4.1],𝑉(𝐻_𝑋) =𝑋_sing.

Definition 3.13. Let𝑋 ⊂ℝ^𝑁be an algebraic subset and𝑒∈ℕ_≥₀. We set L^(𝑒)(𝑋) ={

𝛾 ∈L(𝑋),∃ℎ∈𝐻_𝑋, ℎ(𝛾(𝑡))≢0 mod𝑡^𝑒+1} Remark 3.14. From now on, we set

L(𝑋_sing) ={

𝛾 ∈L(ℝ^𝑁),∀ℎ∈𝐻_𝑋, ℎ(𝛾(𝑡)) = 0} and

L_𝑛(𝑋_sing) ={

𝛾∈L_𝑛(ℝ^𝑁),∀ℎ∈𝐻_𝑋, ℎ(𝛾(𝑡))≡0 mod𝑡^𝑛+1} . Notice that

{𝛾∈L(ℝ^𝑁),∀ℎ∈𝐻_𝑋, ℎ(𝛾(𝑡)) = 0}

={

𝛾 ∈L(ℝ^𝑁),∀𝑓 ∈𝐼(𝑋_sing), 𝑓(𝛾(𝑡)) = 0} but be careful that

{𝛾 ∈L_𝑛(ℝ^𝑁),∀ℎ∈𝐻_𝑋, ℎ(𝛾(𝑡))≡0 mod𝑡^𝑛+1}

≠{

𝛾 ∈L_𝑛(ℝ^𝑁),∀𝑓 ∈𝐼(𝑋_sing), 𝑓(𝛾(𝑡))≡0 mod𝑡^𝑛+1}

Notice also that since the proof of Greenberg Theorem3.9is algebraic, it holds forL(𝑋_sing) (just use the ideal𝐻_𝑋in the proof).

Remark 3.15. L(𝑋) =

⎛⎜

⎜⎝

⋃

𝑒∈ℕ_≥₀

L^(𝑒)(𝑋)

⎞⎟

⎟⎠

⨆L(𝑋_sing)

The following proposition is a real version of [10, Lemma 4.1]. Its proof is quite similar to the one of [8, Lemma 4.5].

Proposition 3.16. Let𝑋be a𝑑-dimensional algebraic subset ofℝ^𝑁and𝑒∈ℕ_≥0. Then, for𝑛≥𝑒, (i) 𝜋_𝑛(

L^(𝑒)(𝑋))

∈AS (ii) 𝜋_𝑛^𝑛+1∶𝜋_𝑛+1(

L^(𝑒)(𝑋))

→𝜋_𝑛(

L^(𝑒)(𝑋))

is anASpiecewise trivial fibration with fiberℝ^𝑑. Proof. By [8, Lemma 4.7],L^(𝑒)(𝑋)is covered by finitely many sets of the form

𝐴_{𝐟,ℎ,𝛿}={

𝛾 ∈L(ℝ^𝑁),(ℎ𝛿)(𝛾(𝑡))≢0 mod𝑡^𝑒+1} where𝐟 = (𝑓₁,…, 𝑓_𝑁−𝑑) ∈𝐼(𝑋)^𝑁−𝑑,𝛿is a𝑁−𝑑minor of the Jacobian matrix

(

𝜕𝑓_𝑖

𝜕𝑥_𝑗

)

𝑖=1,…,𝑁−𝑑 𝑗=1,…,𝑁

andℎ∈ ((𝑓₁,…, 𝑓_𝑁−𝑑) ∶𝐼(𝑋)). Moreover, L(𝑋) ∩𝐴_{𝐟,ℎ,𝛿} ={

𝛾∈L(ℝ^𝑁), 𝑓₁(𝛾(𝑡)) =⋯=𝑓_𝑁−𝑑(𝛾(𝑡)) = 0,(ℎ𝛿)(𝛾(𝑡))≢0 mod𝑡^𝑒+1} ,

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so thatL^(𝑒)(𝑋) =L(𝑋) ∩ ⋃

f inite

𝐴_{𝐟,ℎ,𝛿} = ⋃

f inite

(L(𝑋) ∩𝐴_{𝐟,ℎ,𝛿}) . For𝑒^′≤𝑒, we set

𝐴_{𝐟,ℎ,𝛿,𝑒}′= {

𝛾 ∈𝐴_{𝐟,ℎ,𝛿},ord_𝑡𝛿(𝛾(𝑡)) =𝑒^′,ord_𝑡𝛿^′(𝛾(𝑡))≥𝑒^′,for all𝑁−𝑑minor𝛿^′of (𝜕𝑓_𝑖

𝜕𝑥_𝑗 )}

in order to refine the above cover:L^(𝑒)(𝑋) = ⋃

f inite

(L(𝑋) ∩𝐴_{𝐟,ℎ,𝛿,𝑒}′

).

Fix some set𝐴=𝐴_{𝐟,ℎ,𝛿,𝑒}′∩L(𝑋). Notice that if𝜋_𝑛(𝛾) ∈𝜋_𝑛(𝐴)and if𝜋_𝑛+1(𝜂) ∈𝜋_𝑛+1(L^(𝑒)(𝑋)) is in the preimage of𝜋_𝑛(𝛾)by𝜋_𝑛^𝑛+1then𝜋_𝑛+1(𝜂) ∈𝜋_𝑛+1(𝐴).

Indeed,𝜂 ∈ L(𝑋)so 𝑓₁(𝜂) = ⋯ = 𝑓_𝑁−𝑑(𝜂) = 0and since𝜋_𝑛(𝜂) = 𝜋_𝑛(𝛾), we also get that (ℎ𝛿)(𝜂(𝑡))≢0 mod𝑡^𝑒+1,ord_𝑡𝛿(𝜂(𝑡)) =𝑒^′andord_𝑡𝛿^′(𝜂(𝑡))≥𝑒^′.

Hence it is enough to prove the lemma for𝜋^𝑛+1_𝑛 ∶𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴).

We are first going to prove that the fibers of𝜋_𝑛^𝑛+1 ∶𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴)are𝑑-dimensional affine subspaces ofℝ^𝑁. We can reorder the coordinates so that𝛿is the determinant of the first𝑁−𝑑columns ofΔ =

(

𝜕𝑓_𝑖

𝜕𝑥_𝑗

)

. Then, similarly to the proof of [8, Lemma 4.5], there is a matrix𝑃 such that𝑃Δ = (𝛿𝐼_𝑁−𝑑, 𝑊)and∀𝛾∈𝐴, 𝑊(𝛾(𝑡))≡0 mod𝑡^𝑒^′.

Fix𝛾 ∈𝐴. The elements of the fiber of𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴)over𝜋_𝑛(𝛾),𝛾 ∈𝐴, are exactly the 𝜋_𝑛+1(

𝛾(𝑡) +𝑡^𝑛+1𝜈(𝑡)) for𝜈∈ℝ{𝑡}^𝑑such that𝐟(𝛾(𝑡) +𝑡^𝑛+1𝜈(𝑡)) = 0.

Using Taylor expansion, this last condition becomes

𝐟(𝛾(𝑡)) +𝑡^𝑛+1Δ(𝛾(𝑡))𝜈(𝑡) +𝑡^2(𝑛+1)(⋯) = 0 Or equivalently, since𝛾 ∈𝐴,

𝑡^𝑛+1Δ(𝛾(𝑡))𝜈(𝑡) +𝑡^2(𝑛+1)(⋯) = 0 Multiplying by𝑡^{−𝑛−1−𝑒}^′𝑃, we get

𝑡^−𝑒^′(

𝛿(𝛾(𝑡))𝐼_𝑁−𝑑, 𝑊(𝛾(𝑡)))

𝜈(𝑡) +𝑡^{𝑛+1−𝑒}^′(⋯) = 0

Notice thatord_𝑡(𝛿(𝛾(𝑡)) =𝑒^′. Hence, by Hensel’s lemma and Artin approximation theorem, the sought fiber is the set of

𝜋_𝑛+1( 𝛾(𝑡))

+𝑡^𝑛+1𝜈₀ with𝜈₀satisfying the linear system induced by

𝑡^−𝑒^′(

𝛿(𝛾(𝑡))𝐼_𝑁−𝑑, 𝑊(𝛾(𝑡)))

𝜈₀≡0 mod𝑡

Let𝜈₀be a solution, then its first𝑁−𝑑coefficients are expressed as linear combinations of the last𝑑. Therefore each fiber of𝜋_𝑛^𝑛+1 ∶𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴)is a𝑑-dimensional affine subspace ofℝ^𝑁.

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By Greenberg Theorem3.9, there is a𝑐∈ℕ_≥₀such that𝜋_𝑐𝑛(𝐴)is anAS-set. Then𝜋_𝑛(𝐴) is anAS-set as the image of𝜋^𝑐𝑛_𝑛 ∶𝜋_𝑐𝑛(𝐴)→𝜋_𝑛(𝐴)whose fibers have odd Euler characteristic with compact support, see [40, Theorem 4.3].

Finally, notice that𝜋_𝑛+1(𝐴)⊂ 𝜋_𝑛(𝐴) ×ℝ^𝑁and that𝜋^𝑛+1_𝑛 ∶𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴)is simply the first projection. Then, according to the following lemma,𝜋^𝑛+1_𝑛 ∶𝜋_𝑛+1(𝐴)→𝜋_𝑛(𝐴)is anAS

piecewise trivial fibration. ■

Lemma 3.17. Let𝐴be anAS-set,Ω ⊂ 𝐴×ℝ^𝑁 be anAS-set and𝜋 ∶ Ω → 𝐴be the natural projection.

Assume that for all𝑥∈𝐴, the fiberΩ_𝑥=𝜋⁻¹(𝑥)is a𝑑-dimensional affine subspace ofℝ^𝑁. Then𝜋∶ Ω→𝐴is anASpiecewise trivial fibration.

Proof. Up to embedding the space of𝑑-dimensional affine subspaces ofℝ^𝑁 into the space of𝑑+ 1-dimensional vector suspaces ofℝ^𝑁+1, we may assume that the fibers are linear subspaces.

Denote by𝐺=𝔾_{𝑁 ,𝑑}the Grassmannian of𝑑-dimensional linear subspaces ofℝ^𝑁and let 𝐸→𝐺be the tautological bundle; i.e. for𝑔∈𝐺, the fiber𝐸_𝑔is the subspace given by𝑔.

We are first going to prove that the following set isAS, 𝐴̃={

(𝑥, 𝑔) ∈𝐴×𝐺,Ω_𝑥=𝐸_𝑔} .

Identifying𝐺with the set of symmetric idempotent(𝑁 ×𝑁)-matrices of trace𝑑, see [6, Proof of Theorem 3.4.4], for𝑖 = 1,…, 𝑁 we define the regular map𝑤_𝑖 ∶ 𝐺 → ℝ^𝑁as the projection to the coordinates corresponding to the𝑖th-column of such matrices. Then𝐸_𝑔is linearly spanned by(

𝑤_𝑖(𝑔))

. Hence𝐿_𝑖={

(𝑣, 𝑔) ∈ℝ^𝑁×𝐺, 𝑣=𝑤_𝑖(𝑔)}

isAS. Thus {(𝑥, 𝑣, 𝑔) ∈𝐴×ℝ^𝑁×𝐺, 𝑣=𝑤_𝑖(𝑔) ∈ Ω_𝑥}

= (Ω ×𝐺) ∩ (𝐴×𝐿_𝑖) isASand its projection

𝑋_𝑖={

(𝑥, 𝑔) ∈𝐴×𝐺, 𝑤_𝑖(𝑔) ∈ Ω_𝑥}

is alsoASas the image of anAS-set by an injectiveAS-map, see [40, Theorem 4.5].

Then𝐴̃=⋂

𝑖𝑋_𝑖isASas claimed.

Let𝑥₀∈ 𝐴. Fix a coordinate system onℝ^𝑁 such thatΩ_𝑥

0 = {

𝑥_𝑑+1=⋯=𝑥_𝑁 = 0} and fix the projectionΛ ∶ℝ^𝑁→ℝ^𝑑defined byΛ(𝑥₁,…, 𝑥_𝑁) = (𝑥₁,…, 𝑥_𝑑). Let𝜔∶𝐴̃→ℝ(^𝑁_𝑑)be such that the coordinates of𝜔(𝑥, 𝑔)are the𝑑-minors of(

Λ(𝑤_𝑖(𝑔)))

𝑖=1,…,𝑁. Then 𝐴̃₀={

(𝑥, 𝑔) ∈𝐴,̃ Λ ∶ Ω_𝑥→ℝ^𝑑 is of rank𝑑} is anAS-set as the complement of𝜔⁻¹(0). Therefore

𝐴₀={

𝑥∈𝐴, Λ ∶ Ω_𝑥→ℝ^𝑑 is of rank𝑑}

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isASas the image of theAS-set𝐴̃₀by the projection to the first factor which is an injective AS-map.

ThusΦ(𝑥, 𝑣) = (𝑥,Λ(𝑣))is a bijection whose graph isAS.

𝜋⁻¹(𝐴₀) ^Φ //

𝜋 ##

𝐴₀×ℝ^𝑑

pr_𝐴

{{ 0

𝐴₀

Consequently𝜋 ∶ Ω→𝐴is locally trivial for theAS-topology and hence it is anASpiece-

wise trivial fibration. ■

3.2 The Grothendieck ring of AS -sets

Definition 3.18. Let𝐾₀(AS)be the free abelian group generated by[𝑋],𝑋∈AS, modulo (i) Let𝑋, 𝑌 ∈AS. If there is a bijection𝑋→𝑌 withAS-graph, then[𝑋] = [𝑌];

(ii) If𝑌 ⊂ 𝑋are twoAS-sets, then[𝑋] = [𝑋⧵𝑌] + [𝑌].

We put a ring structure on𝐾₀(AS)by adding the following relation:

(iii) If𝑋, 𝑌 ∈AS, then[𝑋×𝑌] = [𝑋][𝑌].

Notation 3.19. We set0 = [∅],1 = [pt]and𝕃= [ℝ].

Remark 3.20. Notice that0is the unit of the addition and1the unit of the multiplication.

Remark 3.21. If𝑝∶𝐸→𝐵is anASpiecewise trivial fibration with fiber𝐹, then [𝐸] = [𝐵][𝐹]

Definition 3.22. We setM=𝐾₀(AS)[ 𝕃⁻¹]

.

The authors of [31] proved there exists a unique additive (and multiplicative) invariant of real algebraic varieties up to biregular morphisms which coincides with the Poincaré polynomial for compact non-singular varieties. This construction relies on the weak fac- torization theorem. Then G. Fichou [12] extended this construction toAS-sets up Nash isomorphisms.

Next, in [32], they gave a new construction of the virtual Poincaré polynomial, related to the weight filtration they introduced in real algebraic geometry. They proved it is an invariant ofAS-sets up to homeomorphism withAS-graph. Actually, using the additivity, they proved it is an invariant ofAS-sets up toAS-bijections (see [9, Remark 4.15]).

Theorem 3.23([31,12,32]). There is a unique ring morphism𝛽 ∶𝐾₀(AS)→ℤ[𝑢]such that if 𝑋compact and non-singular then

𝛽([𝑋]) =∑

𝑖≥0

dim𝐻_𝑖(𝑋,ℤ₂)𝑢^𝑖. We say that𝛽([𝑋])is thevirtual Poincaré polynomialof𝑋.

Moreover, if𝑋≠∅,deg𝛽(𝑋) = dim𝑋and the leading coefficient of𝛽(𝑋)is positive.

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Theorem 3.24([13, Theorem 1.16]). The virtual Poincaré polynomial is a ring isomorphism 𝛽∶𝐾₀(AS)←←←←←←←→^∼ ℤ[𝑢].

Remark 3.25. The virtual Poincaré polynomial induces a ring isomorphism 𝛽∶M→ℤ[𝑢, 𝑢⁻¹].

Definition 3.26. We define the ring M̂as the completion ofMwith respect to the ring filtration⁵defined by the following subgroups induced by dimension

F^𝑚M=⟨

[𝑆]𝕃^−𝑖, 𝑖− dim𝑆≥𝑚⟩ i.e.

M̂= lim←←←←←←←←←←←M/ F^𝑚M.

Proposition 3.27. The virtual Poincaré polynomial induces a ring isomorphism 𝛽∶M̂→ℤ[𝑢]J𝑢⁻¹K.

Proof. We have to prove that lim←←←←←←←←←←←

𝑚

ℤ[𝑢, 𝑢⁻¹]/

F^𝑚ℤ[𝑢, 𝑢⁻¹] =ℤ[𝑢]J𝑢⁻¹K whereF^𝑚ℤ[𝑢, 𝑢⁻¹] =⟨

𝑓 ∈ℤ[𝑢, 𝑢⁻¹],deg𝑓 ≤−𝑚⟩ For𝑛 < 𝑚, we define .

𝜌_𝑚,𝑛∶ ℤ[𝑢, 𝑢⁻¹]/

F^𝑚ℤ[𝑢, 𝑢⁻¹] → ℤ[𝑢, 𝑢⁻¹]/

F^𝑛ℤ[𝑢, 𝑢⁻¹]

by ∑𝑟

𝑘=−𝑚+1

𝑎_𝑘𝑢^𝑘↦

∑𝑟

𝑘=−𝑛+1

𝑎_𝑘𝑢^𝑘 and

𝜌_𝑚∶ℤ[𝑢]J𝑢⁻¹K→ ℤ[𝑢, 𝑢⁻¹]/

F^𝑛ℤ[𝑢, 𝑢⁻¹]

by ∑𝑟

𝑘=−∞

𝑎_𝑘𝑢^𝑘↦

∑𝑟

𝑘=−𝑚+1

𝑎_𝑘𝑢^𝑘 By construction,

lim←←←←←←←←←←←

𝑚

ℤ[𝑢, 𝑢⁻¹]/

F^𝑚ℤ[𝑢, 𝑢⁻¹] = {

(𝑓_𝑚) ∈ ∏

𝑚∈ℤ

ℤ[𝑢, 𝑢⁻¹]/

F^𝑚ℤ[𝑢, 𝑢⁻¹], 𝑛 < 𝑚⇒𝜌_𝑚,𝑛(𝑓_𝑚) =𝑓_𝑛 }

The morphism

𝜑∶ℤ[𝑢]J𝑢⁻¹K→lim

←←←←←←←←←←←

𝑚

ℤ[𝑢, 𝑢⁻¹]/

F^𝑚ℤ[𝑢, 𝑢⁻¹]

defined by𝑓 ↦(𝜌_𝑚(𝑓))_𝑚∈ℤis an isomorphism. ■

5i.e.F^𝑚+1M⊂F^𝑚MandF^𝑚M⋅F^𝑛M⊂F^𝑚+𝑛M. The last condition induces a ring structure on the group M̂.

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Definition 3.28. For𝛼∈M, we define the virtual dimension of𝛼bydim𝛼=𝑚where𝑚is the only integer such that𝛼∈F^−𝑚M⧵F^−𝑚+1M.

Proposition 3.29. dim𝛼= deg(𝛽(𝛼))

Remark 3.30. Notice that for𝑥∈ M,(𝑥+F^𝑚M)_𝑚defines a basis of open neighborhoods.

This topology coincides with the one induced by the non-archimedean norm‖⋅‖∶M→ℝ defined by‖𝛼‖ = 𝑒^dim(𝛼). The completionM̂is exactly the topological completion with respect to this non-archimedean norm. Particularly,

• Let(𝛼_𝑛) ∈M, then𝛼_𝑛→0inM̂if and only ifdim(𝛼_𝑛)→−∞.

• Let(𝛼_𝑛) ∈M, then∑

𝑛𝛼_𝑛converges inM̂if and only if𝛼_𝑛→0inM̂.

• The following equality holds inM̂:

(1 −𝕃^−𝑝)

∑∞

𝑖=0

𝕃^−𝑝𝑖= 1

Definition 3.31. We define an order onM̂as follows. For𝑎, 𝑏 ∈ M̂, we set𝑎 ⪯ 𝑏if and only if either𝑏= 𝑎or the leading coefficient of the virtual Poincaré polynomial𝛽(𝑏−𝑎)is positive.

Remark 3.32. Notice that this real setting has good algebraic properties compared to its complex counterpart:

• 𝐾₀(AS) is an integral domain whereas 𝐾₀(Var_ℂ)is not [43]. Indeed, there is no zero divisor in𝐾₀(AS)whereas the class of the affine line is a zero divisor of𝐾₀(Var_ℂ)[7] [30].

Notice that in particular𝐾₀(Var_ℂ)→M_ℂ=𝐾₀(Var_ℂ)[ 𝕃⁻¹_ℂ ]

is not injective.

• The natural map M → M̂is injective. Indeed its kernel is ∩_𝑚F^𝑚Mand the virtual Poincaré polynomial allows us to conclude: if𝛼∈ ∩_𝑚F^𝑚M, then, for all𝑚∈ℤ,deg𝛼≤−𝑚 and hence𝛼= 0. In the complex case, it is not known whetherM_ℂ→M̂_ℂis injective.

3.3 Real motivic measure

M. Kontsevitch introduced motivic integration in the non-singular case where the measurable sets were the cylinders by using the fact that they are stable. Still in the non-singular case, V. Batyrev [1, §6] enlarged the collection of measurable sets: a subset of the arc space is measurable if it may be approximated by stable sets.

Concerning the singular case, J. Denef and F. Loeser [10] defined a measure and a first family of measurable sets including cylinders. Then, in [11, Appendix], they used the tools they developped in the singular case to adapt the definition of V. Batyrev to the singular case. See also [28].

From now on we assume that𝑋is a𝑑-dimensional algebraic subset ofℝ^𝑁. Definition 3.33. A subset𝐴 ⊂L(𝑋)is said to bestableat level𝑛if:

• For𝑚≥𝑛,𝜋_𝑚(𝐴)is anAS-subset ofL_𝑚(𝑋);

• For𝑚≥𝑛,𝐴=𝜋_𝑚⁻¹(𝜋_𝑚(𝐴));

• For𝑚≥𝑛,𝜋^𝑚+1_𝑚 ∶𝜋_𝑚+1(𝐴)→𝜋_𝑚(𝐴)is anASpiecewise trivial fibration with fiberℝ^𝑑.

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Remark 3.34. Notice that, for the two first points, it is enough to verify that𝜋_𝑛(𝐴) ∈ASand that𝐴=𝜋⁻¹_𝑛 (𝜋_𝑛(𝐴))only for𝑛. Indeed, then, for𝑚≥𝑛,𝜋_𝑚(𝐴) = (𝜋_𝑛^𝑚)⁻¹(𝜋_𝑛(𝐴))is anAS-set as inverse image of anAS-set by a projection.

Then the following proposition holds (notice that the condition𝐴=𝜋⁻¹_𝑚 (𝜋_𝑚(𝐴))is quite important).

Proposition 3.35. If𝐴, 𝐵are stable subsets ofL(𝑋), then𝐴∪𝐵,𝐴∩𝐵and𝐴⧵𝐵are stable too.

Remark 3.36. Notice thatL(𝑋)may not be stable when𝑋is singular.

Definition 3.37. For𝐴 ⊂L(𝑋)a stable set, we define its measure by 𝜇(𝐴) = [𝜋_𝑛(𝐴)]

𝕃^(𝑛+1)𝑑 ∈M, 𝑛 ≫1.

Definition 3.38. The virtual dimension of a stable set is

dim(𝐴) = dim(𝜋_𝑛(𝐴)) − (𝑛+ 1)𝑑, 𝑛 ≫1.

Remark 3.39. Notice that the previous definitions don’t depend on𝑛for𝑛big enough.

Remark 3.40. Notice thatdim(𝐴) = dim(𝜇(𝐴))where the second dimension is the one introduced in Definition3.28.

Definition 3.41. A subset𝐴 ⊂L(𝑋)is measurable if, for every𝑚∈ℤ_<0, there exist

• a stable set𝐴_𝑚⊂L(𝑋);

• a sequence of stable sets(𝐶_𝑚,𝑖⊂L(𝑋))_𝑖_≥₀ such that

• ∀𝑖,dim𝐶_𝑚,𝑖< 𝑚;

• 𝐴Δ𝐴_𝑚⊂∪𝐶_𝑚,𝑖

Then we define the measure of𝐴by𝜇(𝐴) = lim

𝑚→−∞𝜇(𝐴_𝑚).

Proposition 3.42. The previous limit is well defined inM̂and doesn’t depend on the choices.

The proof of the above Proposition, presented below, relies on the following two lemmas.

Lemma 3.43. Let(𝐴_𝑖)_𝑖∈ℕ

≥0 be a decreasing sequence of non-emptyAS-sets 𝐴₁⊃ 𝐴₂⊃⋯

Then ⋂

𝑖∈𝑁

𝐴_𝑖≠∅.

Proof. Recall that𝐴^AS denotes the smallest closedAS-set containing𝐴. We have the following sequence which stabilizes by noetherianity of theAS-topology:

𝐴₁ÂS ⊃ 𝐴₂ÂS ⊃⋯⊃ 𝐴_𝑘ÂS=𝐴_𝑘+1ÂS=⋯

Recall thatAS-sets are exactly the constructible subsets of projective spaces for theAS- topology whose closed sets are the semialgebraic arc-symmetric sets in the sense of [22].

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Hence 𝐴_𝑙 = ∪_{f inite}(𝑈_𝑖 ∩𝑉_𝑖) where 𝑈_𝑖 isAS-open,𝑉_𝑖 isAS-closed and 𝑈_𝑖 ∩𝑉_𝑖 ≠ ∅. We may assume that the𝑉_𝑖’s are irreducible (up to spliting them) and that𝑈_𝑖∩𝑉_𝑖ÂS =𝑉_𝑖 (up to replacing𝑉_𝑖by𝑈_𝑖∩𝑉_𝑖ÂS). Hence we obtain the following decomposition as a union of finitely many irreducible closed subsets𝐴_𝑘ÂS = ∪𝑉_𝑖 (it is not necessarily the irreducible decomposition since we may have𝑉_𝑖 ⊂ 𝑉_𝑗).

Fix𝑍 anAS-irreducible subset of𝐴_𝑘^AS. By the previous discussion, for𝑙 ≥ 𝑘, there exists𝑈_𝑙an open denseAS-subset of𝑍such that𝑈_𝑙⊂ 𝐴_𝑙.

By [40, Remark 2.7],dim(𝑍⧵𝑈_𝑙)<dim𝑈_𝑙so that𝑍⧵𝑈_𝑙is a closed subset of𝑍with empty interior for the Euclidean topology. From Baire theorem, we deduce that the Euclidean interior of∪_𝑙≥𝑘𝑍⧵𝑈_𝑙is empty. Hence∩_𝑙≥𝑘𝑈_𝑙is non-empty. ■

The following lemma is an adaptation to the real context of [1, Theorem 6.6].

Lemma 3.44. Let𝐴 ⊂L(𝑋)be a stable set and(𝐶_𝑖)_𝑖∈ℕ

≥0 be a family of stable sets such that 𝐴 ⊂ ⋃

𝑖∈ℕ_≥₀

𝐶_𝑖 Then there exists𝑙∈ℕ_≥0such that

𝐴 ⊂

⋃𝑙

𝑖=0

𝐶_𝑖

Proof. Without loss of generality, we may assume that𝐶_𝑖⊂ 𝐴(up to replacing𝐶_𝑖by𝐶_𝑖∩𝐴).

Set𝐷_𝑖=𝐴⧵(

𝐶₁∪⋯∪𝐶_𝑖)

so that we get a decreasing sequence of stable sets 𝐷₁⊃ 𝐷₂⊃ 𝐷₃⊃⋯

satisfying ⋂

𝑖∈ℕ_≥₀

𝐷_𝑖= ∅

Assume by contradiction that𝐴may not be covered by finitely many𝐶_𝑖, then

∀𝑖∈ℕ_≥0, 𝐷_𝑖≠∅

Now assume that𝐴is stable at level𝑛and that𝐷_𝑖is stable at level𝑛_𝑖 ≥𝑛. Then𝜋_𝑛(𝐷_𝑖) = 𝜋_𝑛^𝑛^𝑖(𝜋_𝑛

𝑖(𝐷_𝑖)) ∈ASas the image of anAS-set by a regular map whose fibers have odd Euler characteristic with compact support, see [40, Theorem 4.3]. Hence, by Lemma3.43,

𝐵_𝑛= ⋂

𝑖∈ℕ_≥₀

𝜋_𝑛(𝐷_𝑖)≠∅ Choose𝑢_𝑛∈𝐵_𝑛.

Now set

𝐵_𝑛+1= ⋂

𝑖∈ℕ_≥0

𝜋_𝑛+1(𝐷_𝑖)≠∅

As before each𝜋_𝑛+1(𝐷_𝑖)is a non-emptyAS-set. Notice that(𝜋_𝑛^𝑛+1)⁻¹(𝑢_𝑛)is a non-emptyAS- subset ofL_𝑛+1(𝑋). Then, by Lemma3.43,𝐵_𝑛+1 ∩ (𝜋_𝑛^𝑛+1)⁻¹(𝑢_𝑛) ≠ ∅. This way, there exists 𝑢_𝑛+1∈𝐵_𝑛+1such that𝜋^𝑛+1_𝑛 (𝑢_𝑛+1) =𝑢_𝑛.

Therefore, we may inductively construct a sequence(

𝑢_𝑚∈L_𝑚(𝑋))

𝑚≥𝑛such that:

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets