𝑞
𝑋
𝑓 //𝑌
By choosing the algebraic subset𝑆 ⊂ 𝑋bigger (but still withdim𝑆 < 𝑑) we may assume that(𝑝◦𝜎)∗induces a bijectionL(𝑀)⧵L(𝑆′) →L(𝑋)⧵L(𝑆), where𝑆′ = (𝑝◦𝜎)−1(𝑆). Note thatdim𝑆′< 𝑑. Thus the diagram (2) induces a diagram
L(𝑀)⧵L(𝑆′) iI(𝑝◦𝜎)∗
vvvv
(𝑞◦𝜎)∗
&&
L(𝑋)⧵L(𝑆)
𝑓∗ //L(𝑌)
where we denote𝑓∗= (𝑞◦𝜎)∗◦(𝑝◦𝜎)−1∗ .
Now we show how to compute the order of the Jacobian determinant of𝑓along𝛾using a resolution diagram.
Proposition 4.10. Let𝛾 ∈L(𝑋)⧵L(𝑆)and let̃𝛾 = (𝑝◦𝜎)−1∗ (𝛾). Then (3) ord𝑡jac𝑓(𝛾) = ord𝑡jac𝑞◦𝜎(̃𝛾(𝑡)) − ord𝑡jac𝑝◦𝜎(̃𝛾(𝑡)).
Proof. The result derives from the chain rule which holds outside𝑆. ■ Corollary 4.11. Suppose that𝑓is generically of maximal rank. Then the Jacobian determinant of𝑓 is bounded from above, resp. from below, if and only if at every point of𝑀a local generator ofJ𝑝◦𝜎 divides a local generator ofJ𝑞◦𝜎, resp. a local generator ofJ𝑞◦𝜎divides a local generator ofJ𝑝◦𝜎. Remark 4.12. We deduce from the previous corollary that if one of the conditions of Defi-nition4.8is satisfied for one𝑆, then it holds for every𝑆.
4.3 An inverse mapping theorem
Theorem 4.13. Let𝑓 ∶ (𝑋, 𝑥) →(𝑌 , 𝑦)be a germ of semialgebraic homeomorphism between real algebraic sets. Assume that𝜇L(𝑋)(L(𝑋, 𝑥)) =𝜇L(𝑌)(L(𝑌 , 𝑦)).
If𝑓 is generically arc-analytic and if the Jacobian determinant of𝑓 is bounded from below, then the inverse map𝑓−1 ∶𝑌 → 𝑋is also generically arc-analytic and the Jacobian of𝑓 is bounded from above.
Remark 4.14. Notice that arc-analyticity is an open condition for semialgebraic continuous maps (See [27, Theorem 3.1] where it is not necessary to assume that𝑓 is bounded, up to composing𝑓with a real analytic diffeomorphism𝜑∶ℝ→(−1,1)). Hence, since the above statement is local, it is enough to use real analytic arcs centered at𝑥for the arc-analyticity condition.
The same holds for the boundedness of the Jacobian of𝑓: we assume that the arcs of Definition4.8or Corollary4.11are centered at𝑥.
Proof of Theorem4.13. We have the commutative diagram (2). Notice that𝐸= (𝑝◦𝜎)−1(0)is algebraic since𝑝◦𝜎is regular. By Theorem3.67,
𝜇L(𝑋)(
Since we may lift a real analytic arc non-entirely included in the exceptional locus by𝑝◦𝜎, we have
Since𝜇L(𝑌)(L(𝑌 , 𝑦)) =𝜇L(𝑋)(L(𝑋, 𝑥)), we obtain Since𝑀 is non-singular, we may simply write
(𝜇L(𝑌)( Since the Jacobian determinant𝑓 is bounded from below, using Proposition4.10, we get that each summand of the RHS is positive or zero (in the sense of Definition3.31) because the leading coefficient of the virtual Poincaré polynomial of a non-emptyAS-set is positive:
(𝜇L(𝑌)(
Assume that𝑓is not bounded from above, then at least one of the summand of the RHS is positive so that we obtain a contradiction. This proves that𝑓 is bounded from above.
Furthermore, since the RHS is zero, we obtain that
(4) 𝜇L(𝑌)(
(𝑞◦𝜎)∗(L(𝑀 , 𝐸)))
=𝜇L(𝑌)(L(𝑌 , 𝑦))
We are now going to prove that𝑓−1is generically arc-analytic so that it is blow-Nash.
Assume by contradiction there exists𝛾 ∈ L(𝑌 , 𝑦)not entirely included in𝑓(𝑆) ∪𝑌sing which may not be lifted by𝑞◦𝜎. Nevertheless, by [8, Proposition 2.21],
(𝑞◦𝜎)−1(𝛾(𝑡)) =∑
By assumption(𝑞◦𝜎)−1(𝛾(𝑡))is not analytic so that either these expansions don’t coincide or they have a non-integer exponent.
1. We first treat the latter case. Assume that (𝑞◦𝜎)−1(𝛾(𝑡)) =
Since(𝑞◦𝜎)−1 ∶𝑌 ⧵𝑓(𝑆) →𝑀 is continuous and subanalytic, it is locally Hölder so that there exists𝑁 ∈ ℕ≥0satisfying for all real analytic arc𝜂(𝑡)not entirely included in𝑓(𝑆) ∪𝑌sing, of𝑌. Since𝛾 is not entirely included in the singular set of 𝑌, there exists a unique real analytic arc ̃𝛾on𝑀such that𝛾 =𝜌◦̃𝛾. Let𝑒= ord𝑡jac𝜌(̃𝛾(𝑡))and𝑒′be such that Hence𝑐=𝑑and we obtain a contradiction.
Thus, for all𝛾 ∈L(𝑌 , 𝑦)⧵L(𝑓(𝑆) ∪𝑌sing)there exists̃𝛾 ∈L(𝑀 , 𝐸)such that(𝑞◦𝜎)(̃𝛾(𝑡)) = 𝛾(𝑡). Then𝑓−1(𝛾(𝑡)) = (𝑝◦𝜎)(̃𝛾(𝑡))which is real analytic. Therefore𝑓−1 is generically
arc-analytic and so blow-Nash. ■
Remark 4.15. Notice that, in the above proof, we do not need a homeomorphism𝑓 ∶𝑋→𝑌 but only a homeomorphism of𝑓 ∶ Reg(𝑋)→Reg(𝑌).
Under the assumptions of the previous theorem, we derive the following corollary from Lemma4.4.
Corollary 4.16. Let𝑓 ∶ (𝑋, 𝑥) → (𝑌 , 𝑦)be a semialgebraic homeomorphism germ between real algebraic sets withdim𝑋= dim𝑌. Assume moreover that𝜇L(𝑋)(L(𝑋, 𝑥)) =𝜇L(𝑌)(L(𝑌 , 𝑦)).
If𝑓 is blow-Nash and if the Jacobian determinant of𝑓 is bounded from below, then the inverse𝑓−1 is also blow-Nash and the Jacobian determinant of𝑓 is bounded from above.
Remark 4.17. Notice that in the previous results we don’t assume that𝑋 = 𝑌 contrary to [8, Main Theorem 3.5].
Theorem 4.18. Let𝑓 ∶ (𝑋, 𝑥)→(𝑌 , 𝑦)be a semialgebraic homeomorphism germ between algebraic sets withdim𝑋 = dim𝑌. If𝑓 is generically arc-analytic and if the Jacobian determinant of𝑓 is bounded from below, then𝜇L(𝑋)(L(𝑋, 𝑥)) ⪯𝜇L(𝑌)(L(𝑌 , 𝑦)).
Proof. Following the beginning of the proof of Theorem4.13, we obtain : (𝜇L(𝑌)(L(𝑌 , 𝑦)) −𝜇L(𝑋)(L(𝑋, 𝑥))) ∑
𝑖≥0
𝕃−𝑖
⪰( 𝜇L(𝑌)(
(𝑞◦𝜎)∗(L(𝑀 , 𝐸)))
−𝜇L(𝑋)(L(𝑋, 𝑥))) ∑
𝑖≥0
𝕃−𝑖
=∑
𝑛≥0
( 𝜇L(𝑀)(
𝛾 ∈L(𝑀 , 𝐸),ord𝑡jac𝑞◦𝜎(𝛾(𝑡))≤𝑛)
−𝜇L(𝑀)(
𝛾 ∈L(𝑀 , 𝐸),ord𝑡jac𝑝◦𝜎(𝛾(𝑡))≤𝑛) ) 𝕃−𝑛
⪰ 0
■
5 An inverse mapping theorem for inner-Lipschitz maps
5.1 Inner distance
Let𝑋 be a connected semialgebraic subset ofℝ𝑁 equipped with the standard Euclidean distance. We denote by 𝑑𝑋 theinner (also calledgeodesic) distance in 𝑋. By definition, for𝑝, 𝑞 ∈ 𝑋, the inner distance𝑑𝑋(𝑝, 𝑞) is the infimum over the length of all rectifiable curves joining𝑝to𝑞in𝑋. By [24],𝑑𝑋(𝑝, 𝑞)is the infimum over the length of continuous semialgebraic curves in𝑋joining𝑝and𝑞. It is proven in [24] that𝑑𝑋can be approximated uniformly by subanalytic distances.
We recall some results from [24], based on [23]. Let𝜀 > 0, we say that a connected semialgebraic setΓ⊂ℝ𝑁is𝐾𝜀-regularif for any𝑝, 𝑞∈ Γwe have
𝑑Γ(𝑝, 𝑞)≤(1 +𝜀)|𝑝−𝑞|. We state now a semialgebraic version of [24, Proposition 3].
Proposition 5.1. Let𝑋 ⊂ℝ𝑁be a semialgebraic set and𝜀 >0. Then there exists a finite decompo-sition𝑋=⋃
𝜈∈𝑉 Γ𝜈such that:
1. eachΓ𝜈is a semialgebraic connected analytic submanifold ofℝ𝑁, 2. eachΓ𝜈is𝐾𝜀-regular.
Remark 5.2. Given a finite family of semialgebraic sets𝑋𝑖, 𝑖 ∈𝐼, we can find a decompo-sition satisfying the above conditions and such that for any𝑖∈ 𝐼,𝜈 ∈ 𝑉, we have: either Γ𝜈 ⊂ 𝑋𝑖orΓ𝜈∩𝑋𝑖= ∅.
For a𝐶1map𝑓 ∶ 𝑋′ → ℝ𝑀 defined on a submanifold𝑋′ofℝ𝑁 we denote by𝐷𝑝𝑓 ∶ 𝑇𝑝𝑋→ℝ𝑀 its differential at𝑝∈𝑋′. Then the norm of𝐷𝑝𝑓 is defined by
‖𝐷𝑝𝑓‖= sup{
|𝐷𝑝𝑓(𝑣)|∶ 𝑣∈𝑇𝑝,|𝑣|= 1} .
Lemma 5.3. Assume that𝑓𝜈∶ Γ𝜈→ℝ𝑀 is a𝐶1-map, such that for any𝑝∈ Γ𝜈we have‖𝐷𝑝𝑓𝜈‖≤ 𝐿. Then𝑓𝜈is(1+𝜀)𝐿-Lipschitz with respect to the Euclidean distance, hence it extends continuously onΓ𝜈to a Lipschitz map with the same constant.
Proof. Let𝑝, 𝑞∈ Γ𝜈and𝜀′> 𝜀, then, by [24], there exists a𝐶1-semialgebraic arc𝜆∶ [0,1]→ Γ𝜈such that𝑝=𝜆(0),𝑞=𝜆(1)of the length|𝜆|≤(1 +𝜀′)|𝑝−𝑞|. It follows that
|𝑓𝜈(𝑝) −𝑓𝜈(𝑞)|≤𝐿|𝜆|≤(1 +𝜀′)𝐿|𝑝−𝑞|. We obtain the conclusion passing to the limit𝜀′→𝜀.
Notice that, on any metric space, a Lipschitz mapping extends continuously to the
clo-sure with the same Lipschitz constant. ■
Let𝑋and𝑌 be locally closed connected semialgebraic subsets respectively ofℝ𝑁 and ℝ𝑀. They are equipped with the inner distances𝑑𝑋and𝑑𝑌, respectively. Let
𝑓 ∶𝑋→𝑌
be a continuous semialgebraic map. Then there exists a semialgebraic set𝑋′⊂ 𝑋, which is open and dense in𝑋, such that the connected components of𝑋′are analytic submanifolds of ℝ𝑁, possibly of different dimensions. Moreover𝑓 restricted to each connected component of𝑋′is analytic.
Proposition 5.4. The following conditions are equivalent:
(i) 𝑑𝑌(𝑓(𝑝), 𝑓(𝑞))≤𝐿𝑑𝑋(𝑝, 𝑞)for any𝑝, 𝑞∈𝑋, (ii) ‖𝐷𝑝𝑓‖≤𝐿for any𝑝∈𝑋′.
Proof. The implication5.4.(𝑖)⇒5.4.(𝑖𝑖)is obvious since at a smooth point𝑝∈𝑋, the inner and Euclidean distances are asymptotically equal.
To prove the converse let us fix𝑝, 𝑞 ∈𝑋. For any𝜀 >0there exists a continuous semi-algebraic arc𝜆 ∶ [0,1] →𝑋such that𝑝= 𝜆(0),𝑞= 𝜆(1)of the length|𝜆|≤(1 +𝜀)𝑑𝑋(𝑝, 𝑞).
By Proposition5.1there exists a finite decomposition𝑋′ = ⋃
𝜈∈𝑉 Γ𝜈 into𝐾𝜀-regular semi-algebraic connected analytic submanifolds ofℝ𝑁. Let𝑋′′=⋃
𝜈∈𝑉′Γ𝜈be the union of those Γ𝜈 which are open in𝑋′. Note that𝑋′′is dense in𝑋′. It follows that𝑋 ⊂⋃
𝜈∈𝑉′Γ𝜈. Since the arc𝜆is semialgebraic there exists a finite sequence0 =𝑡0<⋯< 𝑡𝑘 = 1such that each 𝜆([𝑡𝑖, 𝑡𝑖+1])⊂ Γ𝜈 for some𝜈 ∈ 𝑉′. By Lemma5.3the length of𝑓(𝜆([𝑡𝑖, 𝑡𝑖+1]))is bounded by (1 +𝜀)|𝜆([𝑡𝑖, 𝑡𝑖+1])|. Hence
|𝑓(𝜆([0,1]))|=
𝑘−1∑
𝑖=0
|𝑓(𝜆([𝑡𝑖, 𝑡𝑖+1])|≤(1 +𝜀)𝐿
𝑘−1∑
𝑖=0
|𝜆([𝑡𝑖, 𝑡𝑖+1])|≤(1 +𝜀)𝐿|𝜆|.
Thus
𝑑𝑌(𝑓(𝑝), 𝑓(𝑞))≤|𝑓(𝜆([0,1]))|≤(1 +𝜀)𝐿|𝜆|≤(1 +𝜀)2𝐿𝑑𝑋(𝑝, 𝑞)
We conclude by taking the limit as𝜀→0. ■
5.2 An inverse mapping theorem
We suppose now that𝑓 ∶ 𝑋 → 𝑌 satisfies Assumption4.6. Thus it is a blow-Nash map between two real algebraic sets of dimension𝑑. Let𝛾 ∈ L(𝑋)⧵L(𝑆). Let us adapt the notation introduced in the paragraph after Assumption4.6. In particular we assume that the limit of tangent spaces𝑇𝛾(𝑡)𝑋in the Grassmannian𝔾𝑁 ,𝑑is equal toℝ𝑑⊂ℝ𝑁. Then, for every𝑖= 1,…, 𝑑and every𝑗= 1,…, 𝑀
𝜂𝑖,𝑗(𝑡) = 𝜕𝑓𝑗
𝜕𝑥𝑖
is semialgebraic. Thus the order of𝜂𝑖,𝑗(𝑡), as𝑡→ 0+is a well defined rational number (or +∞if𝑓𝑗 vanishes identically on𝛾).
Definition 5.5. The order of the Jacobian matrix of𝑓 along𝛾 is defined as ord𝑡→0+Jac𝑓(𝛾(𝑡)) = min
𝑖,𝑗 {ord𝑡→0+𝜂𝑖,𝑗(𝑡)}.
Remark 5.6. The above notion shouldn’t be confused with the order of the Jacobian deter-minantdefined in Definition4.7.
Remark 5.7. It is likely that𝜂𝑖,𝑗(𝑡)is actually meromorphic and it is not necessary, in the above definition, to restrict to𝑡→0+. We leave it as an open problem.
Definition 5.8. We say thatthe Jacobian matrix of𝑓is bounded from aboveif there is an𝑆such that for every𝛾∈L(𝑋)⧵L(𝑆),ord𝑡→0+Jac𝑓(𝛾(𝑡))≥0.
One may show again that if the above condition is satisfied for one𝑆they are satisfied for every𝑆.
The following result follows from Proposition5.4.
Proposition 5.9. Let𝑓 ∶ (𝑋, 𝑥)→ (𝑌 , 𝑦)be a semialgebraic homeomorphism germ between two real algebraic set germs withdim(𝑋, 𝑥) = dim(𝑌 , 𝑦). Then𝑓 ∶ Reg(𝑋)→Reg(𝑌)is inner Lipschitz iff the Jacobian matrix of𝑓 is bounded from above.
Theorem 5.10. Let𝑓 ∶ (𝑋, 𝑥)→(𝑌 , 𝑦)be a semialgebraic homeomorphism germ between two real algebraic set germs withdim(𝑋, 𝑥) = dim(𝑌 , 𝑦). Assume that𝜇L(𝑋)(L(𝑋, 𝑥)) =𝜇L(𝑌)(L(𝑌 , 𝑦)). If 𝑓 is generically arc-analytic and𝑓−1∶ Reg(𝑌)→Reg(𝑋)is inner Lipschitz, then𝑓−1∶𝑌 →𝑋is also generically arc-analytic and𝑓 ∶ Reg(𝑋)→Reg(𝑌)is inner Lipschitz.
Remark 5.11. Notice that both previous results involve the closure of the regular parts of the algebraic sets. The obtained setsReg(𝑋)andReg(𝑌)do not contain any part of smaller dimension but they still may not be smooth submanifolds.
For instance, for the Whitney umbrella𝑋 = {𝑥2 = 𝑧𝑦2}, Reg(𝑋)consists in the canopy (i.e. the𝑧≥0part of𝑋). ThereforeReg(𝑋)is singular along the half-axis{(0,0, 𝑧), 𝑧≥0}.
However it doesn’t contain the handle of the Whitney umbrella (i.e.{(0,0, 𝑧), 𝑧 <0}) which is a smooth manifold of dimension1whereasdim𝑋= 2.
Proof of Theorem5.10. To simplify the exposition we suppose that𝑋, and hence𝑌 as well, is pure-dimensional. That is𝑋= Reg(𝑋)and𝑌 = Reg(𝑌). The proof in the general case is similar.
First we apply Proposition5.4to𝑓−1. Hence the Jacobian determinant of𝑓−1is bounded from above. Therefore the Jacobian determinant of𝑓is bounded from below and we can ap-ply to𝑓Theorem4.13. This shows that𝑓−1is generically arc-analytic and that the Jacobian determinant of𝑓 is bounded from above and below.
Now we show that the Jacobian matrix of𝑓 is bounded from above. Let𝛾∈L(𝑋)⧵L(𝑆).
We may assume, as explained above, thatℝ𝑑 ⊂ ℝ𝑁 is the limit of tangent spaces𝑇𝛾(𝑡)𝑋. Similarly by considering the limit of𝑇𝑓(𝛾(𝑡))𝑌 we may assume that it equalsℝ𝑑 ⊂ℝ𝑀. Then 𝑦1,…, 𝑦𝑑form a local system of coordinates on𝑌 at every𝑓(𝛾(𝑡)),𝑡≠0. By the assumptions the matrix(𝜕𝑥𝑖
𝜕𝑦𝑗)(𝑓(𝛾(𝑡))is bounded and its determinant is a unit. Therefore, by the cofactor formula, its inverse, that is(𝜕𝑥𝑖
𝜕𝑦𝑗)(𝑓(𝛾(𝑡))is bounded. This shows that𝑓 is inner Lipschitz by
Proposition5.9. ■
Remark 5.12. Notice that, in the above proof, we do not need a homeomorphism𝑓 ∶𝑋→𝑌 but only a homeomorphism of𝑓 ∶ Reg(𝑋)→Reg(𝑌).
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