Definition 3.59. Let𝑋 ⊂ℝ𝑁be an algebraic subset. Let𝐴 ⊂L(𝑋)be a measurable set. Let 𝛼∶𝐴→ℕ≥0∪ {∞}be such that each fiber is measurable and𝜇(𝛼−1(∞)) = 0. We say that 𝕃−𝛼is integrable if the following sequence converges inM̂:
∫𝐴𝕃−𝛼d𝜇=∑
𝑛≥0
𝜇( 𝛼−1(𝑛))
𝕃−𝑛
Definition 3.60. We say that a semialgebraic map𝜎 ∶𝑀 →𝑋between semialgebraic sets isgenerically one-to-oneif there exists a semialgebraic set𝑆 ⊂ 𝑋 satisfyingdim(𝑆)<dim(𝑋), dim(
𝜎−1(𝑆))
<dim(𝑀)and∀𝑝∈𝑋⧵𝑆, #𝜎−1(𝑝) = 1.
Definition 3.61. Let𝜎 ∶ 𝑀 → 𝑋 be a Nash map between a𝑑-dimensional non-singular algebraic set𝑀 and an algebraic subset𝑋 ⊂ℝ𝑁. For a real analytic arc𝛾 ∶ (ℝ,0)→𝑀, we set
ord𝑡jac𝜎(𝛾(𝑡)) = min{
ord𝑡𝛿(𝛾(𝑡)),∀𝛿 𝑑-minor of Jac𝜎} ,
where the Jacobian matrixJac𝜎is defined using a local system of coordinates around𝛾(0)in 𝑀.
The following lemma is a generalization of Denef–Loeser change of variables key lemma [10, Lemma 3.4] to generically one-to-one Nash maps in the real context.
Lemma 3.62([8, Lemma 4.5]). Let𝜎 ∶ 𝑀 → 𝑋be a proper generically one-to-one Nash map where𝑀is a non-singular𝑑-dimensional algebraic subset ofℝ𝑝and𝑋a𝑑-dimensional algebraic subset ofℝ𝑁. For𝑒, 𝑒′∈ℕ≥0and𝑛∈ℕ≥0, set
Δ𝑒,𝑒′ = {
𝛾∈L(𝑀),ord𝑡jac𝜎(𝛾(𝑡)) =𝑒, 𝜎∗(𝛾) ∈L(𝑒′)(𝑋) }
, Δ𝑒,𝑒′,𝑛=𝜋𝑛( Δ𝑒,𝑒′
),
where𝜎∗∶L(𝑀)→L(𝑋)is induced by𝜎. Then for𝑛≥max(2𝑒, 𝑒′)the following holds:
(i) Given𝛾 ∈ Δ𝑒,𝑒′ and𝛿 ∈ L(𝑋)with𝜎∗(𝛾) ≡ 𝛿 mod𝑡𝑛+1 there exists a unique𝜂 ∈ L(𝑀) such that𝜎∗(𝜂) =𝛿and𝜂≡𝛾 mod𝑡𝑛−𝑒+1.
(ii) Let𝛾, 𝜂 ∈ L(𝑀). If𝛾 ∈ Δ𝑒,𝑒′ and𝜎∗(𝛾) ≡ 𝜎∗(𝜂) mod𝑡𝑛+1 then𝛾 ≡ 𝜂 mod𝑡𝑛−𝑒+1 and 𝜂∈ Δ𝑒,𝑒′.
(iii) The setΔ𝑒,𝑒′,𝑛is a union of fibers of𝜎∗𝑛. (iv) 𝜎∗𝑛(Δ𝑒,𝑒′,𝑛)is an AS-set and𝜎∗𝑛|Δ
𝑒,𝑒′,𝑛 ∶ Δ𝑒,𝑒′,𝑛 → 𝜎∗𝑛(Δ𝑒,𝑒′,𝑛)is an AS piecewise trivial fibration with fiberℝ𝑒.
Lemma 3.63. Let𝜎∶𝑋→𝑌 be a Nash map between algebraic sets. If𝐴 ⊂L(𝑌)is a cylinder then 𝜎∗−1(𝐴)⊂L(𝑋)is also a cylinder.
Proof. Assume that𝐴 = 𝜋𝑛−1(𝐶)where𝐶 is anAS-subset ofL𝑛(𝑌). Then we have the fol-lowing commutative diagram:
L(𝑋) 𝜎∗ //
𝜋𝑛
L(𝑌)
𝜋𝑛
L𝑛(𝑋) 𝜎
∗𝑛
//L𝑛(𝑌)
Notice that𝜎∗𝑛is polynomial and thus its graph isASso that the inverse image of anAS-set by𝜎∗𝑛is also anAS-set. Hence𝜎∗−1(𝐴) =𝜋−1𝑛 (𝜎∗𝑛−1(𝐶))where𝜎∗𝑛−1(𝐶)isAS. ■ Proposition 3.64. Let𝜎 ∶ 𝑀 → 𝑋be a proper generically one-to-one Nash map where𝑀 is a non-singular𝑑-dimensional algebraic subset ofℝ𝑝and𝑋a𝑑-dimensional algebraic subset ofℝ𝑁. If𝐴 ⊂L(𝑋)is a measurable subset, then the inverse image𝜎∗−1(𝐴)is also measurable.
Proof. Let
𝑆′=𝜎−1(𝑋sing∪𝑆) ∪ Σ𝜎Zar
where𝑆 ⊂ 𝑋is as in Definition3.60andΣ𝜎is the critical set of𝜎. Notice that the Zariski-closure of a semialgebraic set doesn’t change its dimension. ThereforeL(𝑆′)is a measurable subset ofL(𝑀)with measure0.
Hence𝜎∗−1(𝐴)is measurable if and only if𝜎∗−1(𝐴)⧵L(𝑆′)is measurable and then 𝜇(
𝜎∗−1(𝐴))
=𝜇(
𝜎∗−1(𝐴)⧵L(𝑆′))
Since𝐴is measurable, there exists𝐴𝑚and𝐶𝑚,𝑖as in Definition3.41. Hence for all𝑚∈ ℤ<0,
𝜎∗−1(𝐴)Δ𝜎∗−1(𝐴𝑚)⊂⋃
𝑖
𝜎−1∗ (𝐶𝑚,𝑖) and
(1) (
𝜎∗−1(𝐴)⧵L(𝑆′)) Δ(
𝜎∗−1(𝐴𝑚)⧵L(𝑆′))
⊂⋃
𝑖
(𝜎∗−1(𝐶𝑚,𝑖)⧵L(𝑆′))
By Lemma3.63the sets𝜎∗−1(𝐴𝑚)and𝜎∗−1(𝐶𝑚,𝑖)are cylinders, therefore they are stable sets by Proposition3.55since𝑀is non-singular.
By definition of𝑆′,
L(𝑀)⧵L(𝑆′)⊂⋃
𝑒,𝑒′
Δ𝑒,𝑒′
By Lemma3.44, there exists𝑘such that
L(𝑀)⧵L(𝑆′)⊂ ⋃
𝑒,𝑒′≤𝑘
Δ𝑒,𝑒′
Thus, by Lemma3.62,dim(
𝜎∗−1(𝐶𝑚,𝑖)⧵L(𝑆′))
< 𝑘+𝑚.
This allows one to prove that𝜎∗−1(𝐴)⧵L(𝑆′)is measurable by shifting the index𝑚in
(1). ■
Proposition 3.65. Let𝜎 ∶ 𝑀 → 𝑋 be a proper generically one-to-one Nash map where𝑀 is a non-singular𝑑-dimensional algebraic subset ofℝ𝑝and𝑋a𝑑-dimensional algebraic subset ofℝ𝑁. If𝐴 ⊂L(𝑀)is a measurable subset, then the image𝜎∗(𝐴)is also measurable.
Proof. We use the same𝑆′as in the proof of Proposition3.64. ThenL(𝑆′)and𝜎∗( L(𝑆′)) have measure0so that it is enough to prove that𝜎∗(
𝐴⧵L(𝑆′))
is measurable.
Lemma 3.66. There exists𝑘such that for every stable set𝐵 ⊂ L(𝑀)⧵L(𝑆′),𝜎∗(𝐵)is stable and dim(
𝜎∗(𝐵))
<dim(𝐵) −𝑘.
Proof. By definition of𝑆′and Lemma3.44, there exists𝑘such that 𝐵 ⊂L(𝑀)⧵L(𝑆′)⊂ ⋃
𝑒,𝑒′≤𝑘
Δ𝑒,𝑒′
Then the lemma derives from Lemma3.62. ■
Assume that𝐴is measurable with the data𝐴𝑚, 𝐶𝑚,𝑖then
𝐴Δ𝐴𝑚⊂⋃ 𝐶𝑚,𝑖 so that
(𝐴⧵L(𝑆′))Δ(𝐴𝑚⧵L(𝑆′))⊂⋃
𝐶𝑚,𝑖⧵L(𝑆′) and
𝜎∗(𝐴⧵L(𝑆′))Δ𝜎∗(𝐴𝑚⧵L(𝑆′))⊂ 𝜎∗(
(𝐴⧵L(𝑆′))Δ(𝐴𝑚⧵L(𝑆′)))
⊂⋃ 𝜎∗(
𝐶𝑚,𝑖⧵L(𝑆′))
Then we may conclude using Lemma3.66. ■
Theorem 3.67. Let𝜎 ∶𝑀 →𝑋be a proper generically one-to-one Nash map where𝑀is a non-singular𝑑-dimensional algebraic subset ofℝ𝑝and𝑋a𝑑-dimensional algebraic subset ofℝ𝑁. Let𝐴 ⊂L(𝑋)be a measurable set. Let𝛼∶𝐴→ℕ≥0∪ {∞}be such that𝕃−𝛼is integrable.
Then𝕃−(𝛼◦𝜎∗+ord𝑡jac𝜎)is integrable on𝜎∗−1(𝐴)and
∫𝐴∩Im(𝜎∗)
𝕃−𝛼d𝜇L(𝑋)=
∫𝜎−1∗ (𝐴)
𝕃−(𝛼◦𝜎∗+ord𝑡jac𝜎)d𝜇L(𝑀)
where𝜎∗∶L(𝑀)→L(𝑋)is induced by𝜎.
Proof. Set𝛽 =𝛼◦𝜎∗+ ord𝑡jac𝜎. By Proposition3.64,𝜎∗−1(𝐴)and the fibers of𝛼◦𝜎∗are mea-surable.
Notice that
𝛽−1(𝑛) =
⨆𝑛
𝑒=0
((𝛼◦𝜎∗)−1(𝑛−𝑒) ∩ (ord𝑡jac𝜎)−1(𝑒) ∩𝜎∗−1(𝐴))
so that the fibers of𝛽are measurable.
As in the proof of Proposition3.64, up to replacing𝜎∗−1(𝐴)by𝜎∗−1(𝐴)⧵L(𝑆′), we may assume that
𝜎∗−1(𝐴)⊂ ⋃
𝑒,𝑒′≤𝑘
Δ𝑒,𝑒′
Using Lemma3.62, we obtain
Notice thatIm(𝜎∗)is measurable by Proposition3.65. ■
4 An inverse mapping theorem for blow-Nash maps
4.1 Blow-Nash and generically arc-analytic maps
Definition 4.1([22, Définition 4.1]). Let𝑋 and𝑌 be two real algebraic sets. We say that 𝑓 ∶ 𝑋 → 𝑌 is arc-analytic if for every real analytic arc𝛾 ∶ (−1,1) → 𝑋 the composition 𝑓◦𝛾 ∶ (−1,1)→𝑌 is also real analytic.
Definition 4.2([8, Definition 2.22]). Let𝑋and𝑌 be two algebraic sets. We say that the map 𝑓 ∶𝑋 → 𝑌 is generically arc-analytic if there exists an algebraic subset𝑆 ⊂ 𝑋satisfying dim𝑆 <dim𝑋and such that if𝛾 ∶ (−1,1)→𝑋is a real analytic arc not entirely included in 𝑆, then the composition𝑓◦𝛾 ∶ (−1,1)→𝑌 is also real analytic.
Definition 4.3. Let𝑋and𝑌 be two algebraic sets. We say that𝑓 ∶𝑋 →𝑌 is blow-Nash if 𝑓 is semialgebraic and if there exists a finite sequence of algebraic blowings-up with non-singular centers𝜎∶𝑀→𝑋such that𝑓◦𝜎∶𝑀→𝑌 is real analytic (and hence Nash).
Lemma 4.4([8, Lemma 2.27]). Let𝑓 ∶𝑋→𝑌 be a semialgebraic map between two real algebraic sets. Then𝑓 ∶𝑋→𝑌 is blow-Nash if and only if𝑓 is generically arc-analytic.
Remark 4.5. In the non-singular case, the previous lemma derives from [2] or [39].
Assumption 4.6. For the rest of this section we assume that𝑋 ⊂ℝ𝑁and𝑌 ⊂ℝ𝑀 are two 𝑑-dimensional algebraic sets and that𝑓 ∶ 𝑋 → 𝑌 is blow-Nash. Since𝑓 is, in particular, semialgebraic, it is real analytic in the complement of an algebraic subset𝑆of𝑋of dimen-sion< 𝑑. We may choose𝑆sufficiently big so that𝑆 contains the singular set of𝑋 and
the non-analyticity set of𝑓. Because𝑓 is blow-Nash we may suppose, moreover, that𝑓 is analytic on every analytic arc𝛾not included entirely in𝑆. Then for every𝛾 ∈L(𝑋)⧵L(𝑆), 𝑓◦𝛾 ∈L(𝑌).
We say that such𝑓 is generically of maximal rankif the Jacobian matrix of𝑓 is of rank𝑑 on a dense semialgebraic subset of𝑋⧵𝑆.
Let𝛾 ∈L(𝑋)⧵L(𝑆). Then the limit of tangent spaces𝑇𝛾(𝑡)𝑋exists in the Grassmannian 𝔾𝑁 ,𝑑of𝑑-dimensional linear subspaces ofℝ𝑁. After a linear change of coordinates we may assume that this limit is equal toℝ𝑑 ⊂ℝ𝑁. Then(𝑥1,…, 𝑥𝑑)is a local system of coordinates at every𝛾(𝑡),𝑡≠0. Fix𝐽 = {𝑗1,…, 𝑗𝑑)with1≤𝑗1<⋯< 𝑗𝑑≤𝑀. Then, for𝑡≠0,
d𝑓𝑗
1∧⋯∧ d𝑓𝑗
𝑑(𝛾(𝑡)) =𝜂𝐽(𝑡) d𝑥1∧⋯∧ d𝑥𝑑,
where 𝜂𝐽(𝑡) is a semialgebraic function, well-defined for𝑡 ≠ 0. Indeed, let Γ𝑓 ⊂ ℝ𝑁+𝑀 denote the graph of𝑓and let𝜏Γ
𝑓 ∶ Reg(Γ𝑓)→𝔾𝑁+𝑀 ,𝑑be the Gauss map. It is semialgebraic, see e.g. [6, Proposition 3.4.7], [23]. Denote byΓ̃𝑓 the closure of its image and by𝜋𝑓 ∶Γ̃𝑓 → Γ𝑓 the induced projection. Then𝛾 lifts to a semialgebraic arc𝛾 inΓ̃𝑓. The limitslim𝑡→0+𝛾(𝑡) andlim𝑡→0−𝛾(𝑡)exist, and as follows from Proposition4.10they coincide.
Denote by𝐸 → 𝔾𝑁+𝑀 ,𝑑 the tautological bundle. Thus each fiber of𝐸 → 𝔾𝑁+𝑀 ,𝑑 is a𝑑-dimensional vector subspace ofℝ𝑁+𝑀. We denote by(𝑥1,…, 𝑥𝑁, 𝑓1,…𝑓𝑀)the linear coordinates inℝ𝑁+𝑀. Then the restriction of alternating𝑑-forms to each𝑉𝑑 ∈ 𝔾𝑁+𝑀 ,𝑑 gives an identity
d𝑓𝑗
1 ∧⋯∧ d𝑓𝑗
𝑑 =𝜂𝐽(𝑉𝑑) d𝑥1∧⋯∧ d𝑥𝑑
that defines a semialgebraic function𝜂𝐽(𝑉𝑑) on𝔾𝑁+𝑀 ,𝑑 with values in ℝ∪ {±∞}. Then 𝜂𝐽(𝑡) =𝜂𝐽(𝛾(𝑡)). As follows from Proposition4.10,𝜂𝐽(𝑡)is meromorphic andord𝑡𝜂𝐽 ∈ℤ∪ {∞}.The following notion generalizes the order defined in Definition3.61.
Definition 4.7. The order of the Jacobian determinant of𝑓 along𝛾 is defined as ord𝑡jac𝑓(𝛾) = min
𝐽 {ord𝑡𝜂𝐽(𝑡)}.
If𝜂(𝑡)≡0then we define its order as+∞.
Definition 4.8. We say thatthe Jacobian determinant of𝑓 is bounded from above (resp. below) if there exists𝑆 ⊂ 𝑋as in4.6such that for every𝛾 ∈ L(𝑋)⧵L(𝑆),ord𝑡jac𝑓(𝛾) ≥ 0(resp.
ord𝑡jac𝑓(𝛾)≤0).