Semi-Algebraic Functions with Non-Compact Critical Set

Let X⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subset {\mathbb {R}}^n$$\end{document} be a closed semi-algebraic set, F:Rn→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$$\end{document} be a C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}^2$$\end{document} semi-algebraic function and f=F|X:X→Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=F_{|X}:X\rightarrow {\mathbb {R}}^n$$\end{document} be the restriction of F to X. We define the global index of a critical value ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} of f and prove an index formula for χ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi (X)$$\end{document} that generalizes a result previously proved by the authors for the case of isolated critical points. We define also new indices at infinity and prove an alternative index formula for χ(X).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi (X).$$\end{document}


Introduction
Let f : (R n , 0) → (R, 0) be an analytic function germ with an isolated critical point at 0. The Khimshiashvili formula (see Khimshiashvili 1977) states that where 0 < |δ| 1, B is the closed ball of radius centered at 0, ∇ f is the gradient of f and deg 0 ∇ f is the topological degree of the mapping ∇ f |∇ f | : S → S n−1 .
with g − = − f − ω d , g + = f − ω d , ω(x) = x 2 1 + · · · + x 2 n and d is an integer big enough. Sekalski (2005) gives a global counterpart of Khimshiasvili's formula for a polynomial function f : R 2 → R with a finite number of critical points. He considers the set f = {λ 1 , . . . , λ k } of critical values of f at infinity, where λ 1 < λ 2 < · · · < λ k , and its complement R \ f = ∪ k i=0 ]λ i , λ i+1 [ where λ 0 = −∞ and λ k+1 = +∞. Denoting by r ∞ (g) the number of real branches at infinity of a curve {g = 0} in R 2 , he proves that where for i = 0, . . . , k, λ + i is an element of ]λ i , λ i+1 [ and deg ∞ ∇ f is the topological degree of the mapping ∇ f ∇ f : S R → S n−1 , R 1. Gwoździewicz (2009) gives a topological proof of Sekalski's result using Euler integration. He proves that where χ c denotes the Euler characteristic with compact support that we will define later.
The first named author generalizes Sekalski result in Dutertre (2012) by considering a closed semi-algebraic set X ⊂ R n and a C 2 semi-algebraic function f : R n → R such that f |X has a finite number of critical points. In Dutertre and Moya Perez (2016) the authors recover the first named author's results using Euler integration, which clearly simplifies the proofs.
Finally, in , Sect. 3, Araujo, Chen, Andrade and the first named author gave a generalization of the results of Dutertre (2012) when X = R n and f is a semi-tame function with non-isolated critical points, by adapting to the global case the method developed by Szafraniec (1986).
The aim of this paper is to extend these results to the general case, i.e., without any assumption on the set of critical points of the function. We work in the following setting: X ⊂ R n is a closed semi-algebraic set, F : R n → R is a C 2 semi-algebraic function and f = F |X : X → R n is the restriction of F to X .
In Sect. 3, we define the global index of a critical value c i of f , where R c i 1 and 0 < α 1 R c i . Then, we generalize Theorem 3.16 of Dutertre (2012) and Theorem 5.1 of Dutertre and Moya Perez (2016) for the case of a noncompact critical set, that is, we prove that (Theorem 3.2) Using the same techniques we generalize the other results of Dutertre (2012) and Dutertre and Moya Perez (2016) for χ(X ) for the case of a non-compact critical set. As an application, we obtain an index formula for the quotient of two semi-algebraic functions.
In Sect. 4 we define two new indices, the right index at infinity of an asymptotic non-ρ-regular value d i (see Definition 2.15), where R d i 1 and 0 < α We compute these indices in particular cases and we finish the section with a formula that relates χ(X ) with them (Theorem 4.4). This formula can be viewed as a generalization of Corollary 2.3 applied to f : X → R.
We end the paper in Sect. 5 with some real and global Lê-Iomdine type formulas. Namely, by adding or substracting to f a big power of an adapted function, we construct two functions g + and g − that have compact sets of critical points and then we prove that the sum of the global indices of f (respectively − f ) and g − (respectively g + ) coincide (Theorem 5.11). Such results were prevoiusly proved in , when X = R n and f is a semi-tame function.
Let us finish this introduction with a comment. It seems that all these results can be extended to the case of arbitrary real closed fields. Indeed the tools and results that we use (Euler characteristic with compact support, Hardt's theorem, constructible functions, first Thom-Mather's isotopy lemma...) have versions in this case. For instance, Coste and Shiota (1995) proved a version of the Thom-Mather isotopy lemma without integrating vector fields. But in order to do this, one needs to check many (hidden) details.
The authors are grateful to the referee for suggesting valuable improvements.

Euler Integration
Let X ⊂ R n be a semi-algebraic set. We can write it in the following way: and we call it the Euler characteristic with compact support of X . Let us remark that if X is compact, then χ c (X ) = χ(X ). A constructible function ϕ : X → Z is a Z-valued function that can be written as a finite sum If ϕ is a constructible function, the Euler integral of ϕ is defined as Definition 2.1 Let f : X → Y be a continuous semi-algebraic map and let ϕ : X → Z be a constructible function. The push forward f * ϕ of ϕ along f is the function

Link at Infinity and Adapted Radius
For any closed semi-algebraic set equipped with a Whitney stratification X = α∈A S α , we denote by Lk ∞ (X ) the link at infinity of X . It is defined as follows. Let ω : R n → R be a C 2 proper semi-algebraic positive function. Since ω |X is proper, the set of critical points of ω |X (in the stratified sense) is compact. Hence for R sufficiently big, the map ω : X ∩ ω −1 ([R, +∞[) → R is a stratified submersion. The link at infinity of X is the fibre of this submersion. The topological type of Lk ∞ (X ) does not depend on the choice of the function ω (for instance, see Dutertre (2012), Sect. 3).

Definition 2.4
We will say that R > 0 is an adapted radius for X if D : X ∩ D −1 ([R, +∞[) → R is a stratified submersion, where D is the euclidean norm.

Stratified Critical Points and Values
Let us consider from now on a closed semi-algebraic set X ⊂ R n . It is equipped with a finite semi-algebraic Whitney stratification X = a∈A S a . Let F : R n → R be a C 2 -semi-algebraic function and let f = F |X .
where S( p) is the stratum that contains p.
(2) A point c ∈ R is a critical value if there exists p ∈ f −1 (c) such that p is a critical point of f .
(3) If p is an isolated critical point of f , we define the index of f at p by where 0 < δ 1. Let us notice that if X = R n , by Khimshiashvili (1977), ind( f , X , p) = deg p ∇ f .

Lemma 2.7
The set of critical points of f , f , is a closed semi-algebraic subset of X and its set of critical values, f , is finite.

Proof
To prove that f is closed we use Whitney's condition (a) and to prove that f is finite we use the Bertini-Sard Theorem ( Bochnak et al. (1998)).
The following result gives a relation between the Euler characteristic of X and the indices of the p i 's, when X is compact.
Theorem 2.8 (Dutertre (2012), Theorem 3.1) If X is compact and f has a finite number of critical points p 1 , . . . , p l , we have Now, we give some lemmas that we will use later on. For the proofs we refer to Dutertre (2012). We assume that f has a finite number of critical points p 1 , p 2 , . . . , p l .

Lemma 2.9 If δ < 0 is a small regular value of f and R
1 is such that f −1 (0) ∩ B R is a retract by deformation of f −1 (0), then Lemma 2.10 If f is proper then for any α ∈ R, we have We state a Mayer-Vietoris type result that we will apply several times in the paper.

Lemma 2.11
For any α ∈ R, we have Proof By the additivity of χ c , we know that . It is enough to apply the compact case and the relation between χ and χ c .
The following lemma is a consequence of Lemma 2.10 and Lemma 2.11.

Lemma 2.12 If f is proper then for α and α with α < α , we have
Let g : R n → R be a C 2 -semi-algebraic function such that g −1 (0) intersects X transversally. Let us suppose that f |X ∩{g≤0} admits an isolated critical point p in X ∩ {g = 0} which is not a critical point of f . We say that such a point is a correct critical point. If S denotes the stratum of X that contains p, this implies that Remark 2.14 As a consequence of the last lemma and the definition of the index of a critical point p, we get that

Asymptotic Non--Regular Values
Let ρ(x) = 1 + 1 2 (x 2 1 + · · · + x 2 n ). Note that ∇ρ(x) = x, ρ(x) ≥ 1 and the levels of ρ are the spheres of radius greater than or equal to 1. Let f ,ρ be the polar set where S is the stratum that contains x. We have f ⊂ f ,ρ .

Definition 2.15
The set of asymptotic non-ρ-regular values of f is the set defined as follows: The set f was introduced and studied by Tibȃr (1999) when X = R n and f : R n → R is a polynomial. By Lemma 2.2 in Dutertre (2012), we can assume that

Some Others Sets of Special Values
We define four sets of special values. They are values where some changes in the topology of the fibres of f may occur.
are constant in a neighborhood of α.
Proof The first point is proved in Dutertre (2012). Let α / ∈ B( f ) and let α − < α be a value close enough to α. Let R α (resp. R α − ) be an adapted radius for f −1 (α) (resp. f −1 (α − )). We can choose them in such a way that they are also adapted to { f ≤ α} and { f ≤ α − } respectively. The critical points of f |{α − < f <α}∩B R α − can only lie on S R α − , and they point outwards. By Lemma 2.12, this implies that Similarly, we can consider the critical points of − f |{α − < f <α}∩B R α − . Applying Lemma 2.12 twice, we obtain that The same proof works for α + > α, a value close enough to α.

Remark 2.18
Taking into account Proposition 2.17 and basic properties of

Formulas for the Euler Characteristic of a Closed Semi-Algebraic Set in the General Case
Let X be a closed semi-algebraic set, equipped with a finite semi-algebraic Whitney stratification X = a∈A S a . Let F : R n → R be a C 2 semi-algebraic function. We call f = F |X , the restriction of F to X . Let f = {c 1 , c 2 , . . . , c k } be the set of critical values of f .
Let c i be a critical value of f . The partition f −1 (c i ) = a∈A f −1 (c i ) ∩ S a may not be a Whitney stratification, but since Whitney conditions are stratifying, we can refine it in order to get a Whitney stratification is still a Whitney stratification of X .
Definition 3.1 We define the index of a critical value c i of f as with 0 < α 1 and R c i is an adapted radius for f −1 (c i ).

Theorem 3.2 We have
Proof By Hardt's theorem Hardt (1975), there exists a finite set f ⊂ R such that over each connected component of R \ f , f is a semi-algebraic trivial fibration. Let us write To compute the right-hand side of the above equality, we work with each difference As explained above, we can assume that f −1 (b) is a union of strata of our stratification. If R b is sufficiently big and b − is sufficiently close to b, then the (stratified) critical Moreover they are correct and points outwards (Fig. 1).
Therefore, by Lemmas 2.12 and 2.13, we have applying Lemma 2.11 and the definition of the link at infinity. Let us compute Applying the same argument as above, considering the function f |{R b ≤|x|≤R b − } and applying Lemmas 2.12 and 2.13, we obtain that By Lemma 2.11 and the deformation retract argument, we get that Moreover if we chooseb close enough to b, then the intersection is empty (see Fig. 2). This implies that Finally we obtain that Comparing the two expressions for Then we can write Finally we obtain and so Let R b + l be an adapted radius for f −1 (b + l ). We can write obtaining the desired result.
Proof Let b i be a critical value such that f −1 (b i ) has a finite number of singularities p 1 , . . . , p r i . By Lemma 2.9, we know that

Corollary 3.4 We have
Proof By replacing f by − f and applying an analogous procedure as in the last theorem, we arrive to the desired result.

Corollary 3.5 We have
Proof It follows from Theorem 3.2 and Corollary 3.4 by applying Lemma 2.11.

Corollary 3.7 We have
Proof We have arriving to the desired result.

Corollary 3.8 We have
Proof By replacing f by − f and applying an analogous procedure as in the last corollary, we arrive to the desired result.

Corollary 3.9 We have
Proof It follows from the last two corollaries by applying Lemma 2.11.

Remark 3.10 Since ind
Application 3.11 Let us apply these results to the case of a function given as the quotient of two semi-algebraic functions. Let f , g : X → R be two semi-algebraic functions, where X a closed semi-algebraic set and f (resp. g) is the restriction to X of a C 2 semi-algebraic function F (resp. G). We consider their quotient φ := f /g : X \ V (g) → R which is also a semi-algebraic function. Let Y be the following closed semi-algebraic set: We cannot apply Corollary 3.9 since φ is not defined in X , so we work with Y to obtain a formula for the sum of the global indices of the function φ.
Let π : Y → R be the linear function defined by π(x, y) = y. By applying Corollary 3.9, we have that We have that, if t = 0, and so, When t = 0, we have that and so, Let us study the global index of π at the non-zero critical value t. We recall that where R t is an adapted radius for π −1 (t) and 0 < α If R t is big enough and α small enough, then R = R 2 t − (t − α) 2 is an adapted radius for {φ = t} and { f = g = 0}. Therefore we have Therefore, we get We have that and so, Finally we obtain that If furthermore we assume that 0 is a regular value (in the stratified sense) of f , then 0 is a regular value of π and so Taking f = 1, we obtain an index formula for the Euler characteristic of the non-closed semi-algebraic set X \V (g). Namely we have

New Indices at Infinity
By Proposition 2.17, there exists a finite set {e 1 , e 2 , . . . , e s }, e 1 < e 2 < · · · < e s , such that the function t → χ( f −1 (t)) is locally constant on R \ {e 1 , e 2 , . . . , e s }. When X is compact, by Corollary 2.3, we have is empty for t < e 1 and t > e s . The aim of this section is to generalize this equality when X is only closed, by introducing new indices at infinity and applying the results of Sect. 3.
We recall that f is defined by and that it is a finite set {d 1 , d 2 . . . , d m }, with d 1 < d 2 < · · · < d m .
Definition 4.1 We define the right index at infinity of d i as

Fig. 3 The Broughton polynomial
Analogously, we define the left index at infinity of d i as Example 4.2 Let us consider the Broughton polynomial f (x, y) = y(x y − 1) defined on X = R 2 .
Proof We recall that First of all, note that, by the definition of the indices at infinity, By Corollary 3.9, we have By definition, Therefore, we have To conclude, we remark that

Relations with Functions with Compact Critical Set
If f : (C n , 0) → (C, 0) is an analytic function germ with a one-dimensional singular locus then f + l d has an isolated singularity at the origin, where l is a generic linear We get the result.
Proof Let us study first the case when α belongs to an interval of R\ f bounded from above. We can assume that 0 ∈ f and that b < 0 is the greatest negative element of { f (x n )} tend to b as well. As a consequence, there exists R 0 1 such that for all R ≥ R 0 and x ∈ S R ∩ g − ,ρ ∩ {g − ≤ 1 2 α}, f (x) ≤ b+α 2 and g − (x) ≤ b+α 2 . To conclude, we have that Lk Similarly if α belongs to the interval of R\ f not bounded from above, we can suppose that 0 is the biggest bifurcation value and that α > 0. The proof is the same, replacing {g − ≤ b+α 2 } with {g − ≥ α 2 } and taking R such that α + 1 R k < 2α.

Corollary 5.10 We have
We are in position to state the main theorem of this section.
If X = R n , we have that Moreover, if W − is the vector field defined by W − = ρ k+1 ∇ f +∇ρ, then deg ∞ W − = deg ∞ ∇g − and so, We can apply the same procedure to g + and obtain a vector field W + . We note if f is a polynomial then W − and W + are polynomial vector fields.
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