DEGREE FORMULAS FOR THE EULER CHARACTERISTIC OF SOME SEMIALGEBRAIC SETS

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DEGREE FORMULAS FOR THE EULER

CHARACTERISTIC OF SOME SEMIALGEBRAIC SETS

Julie Lapebie

To cite this version:

Julie Lapebie. DEGREE FORMULAS FOR THE EULER CHARACTERISTIC OF SOME SEMI- ALGEBRAIC SETS. 2018. �hal-01829131�

DEGREE FORMULAS FOR THE EULER CHARACTERISTIC OF SOME SEMIALGEBRAIC SETS

JULIE LAPÉBIE

Abstract. We are interested in computing alternate sums of Euler characteristics of some particular semialgebraic sets, intersections of an algebraic one, smooth or with nitely many singularities, with sets given by just one polynomial inequality.

We state theorems relating these alternate sums of characteristics to some topological degrees at innity of polynomial mappings.

1. Introduction

Let F = (F1, . . . , Fk) : Rⁿ → R^k and G = (G1, . . . , Gl) : Rⁿ → R^l be polynomial maps with k, l≥1and let W be F⁻¹(0). We are interested in the semialgebraic sets

W_G(ε) :=W ∩ {(−1)^ε1G₁ ≥0} ∩ · · · ∩ {(−1)^εlG_l≥0}, where ε= (ε₁, . . . , ε_l)∈ {0,1}^l, and we want to express the alternate sum

X

ε∈{0,1}^l

(−1)^|ε|χ(W_G(ε))

in terms of the topological degree at innity of polynomial mappings involving F and G, with|ε|=

l

X

i=1

ε_i.

These semialgebraic sets are natural generalizations of those studied by Szafraniec in [5] and [6]. They are simply dened by equalities and inequalities. If we want to use Morse theory for manifold with corners, they are the natural objects to study.

Moreover when W =Rⁿ, these sets are basics semialgebraic sets.

To give an idea, when l = 1, this alternate sum is equal to (−1)^|0|χ W ∩ {(−1)⁰G₁ ≥0}

+ (−1)^|1|χ(W ∩ {(−1)¹G₁ ≥0})

=χ W ∩ {G₁ ≥0}

−χ W ∩ {G₁ ≤0}

;

2010 Mathematics Subject Classication. 14P10, 14P25, 58K05.

Key words and phrases. Morse theory, manifold with corners, Euler characteristic, semialgebraic sets.

1

and when l= 2, it is equal to

(−1)^|(0,0)|χ W ∩ {(−1)⁰G₁ ≥0} ∩ {(−1)⁰G₂ ≥0}

+(−1)^|(1,0)|χ(W ∩ {(−1)¹G₁ ≥0} ∩ {(−1)⁰G₂ ≥0}) +(−1)^|(1,1)|χ(W ∩ {(−1)¹G₁ ≥0} ∩ {(−1)¹G₂ ≥0}) +(−1)^|(0,1)|χ W ∩ {(−1)⁰G₁ ≥0} ∩ {(−1)¹G₂ ≥0}

=χ W ∩ {G₁ ≥0} ∩ {G₂ ≥0}

−χ(W ∩ {G₁ ≤0} ∩ {G₂ ≥0}) +χ(W ∩ {G₁ ≤0} ∩ {G₂ ≤0})−χ W ∩ {G₁ ≥0} ∩ {G₂ ≤0}

.

There are already several results relating the Euler characteristic of an algebraic set to the topological degree of a polynomial mapping. First, whenW is compact, Bruce [1] and Szafraniec [5] proved that there exists a polynomial P : Rⁿ⁺¹ → R with an isolated critical point at the origin such that

χ(W) = 1

2 (−1)ⁿ−deg₀∇P ,

where deg₀∇P is the topological degree at the origin of the gradient of P.

In all this paper, we will denote by deg_∞φ the topological degree at innity of φ kφk : S_R^N →S₁^N for any mapping φ:R^N+1 →R^N+1 such that φ⁻¹(0) $B_R^N+1.

When k < n and W is a smooth (n−k)-dimensional manifold, Szafraniec constructs in [6] a polynomial mapH :R^n+k →R^n+k such thatH⁻¹(0) is compact and such that χ(W) = (−1)^kdeg_∞H.

In [2], Dutertre studies semialgebraic sets with isolated singularities. Fromf :Rⁿ→R a polynomial function with isolated critical points such that f(0) > 0, he constructs four polynomial mappings H, L, L₁ and L₂ in terms of f and ∇f such that H⁻¹(0), L⁻¹(0), L⁻¹₁ (0) and L⁻¹₂ (0) are compact. Then he gets the equalities:

• if n is even, then

χ(f⁻¹(0)) = deg_∞H+ deg_∞∇f−deg_∞L₂ χ({f ≥0})−χ({f ≤0}) = 1−deg_∞L+ deg_∞L₁.

• If n is odd, then

χ(f⁻¹(0)) = deg_∞L−deg_∞L₁

χ({f ≥0})−χ({f ≤0}) = 1−deg_∞H−deg_∞∇f+ deg_∞L₂.

The aim of this paper is to obtain a similar result as Szafraniec's one in [6] for our sum Pl

i=1(−1)^|ε|χ(W_G(ε)). Then we generalize the results of Dutertre in [2] to the sets W_G(ε)∩ {f ≥0}, W_G(ε)∩ {f ≤0} andW_G(ε)∩ {f = 0}, when W =Rⁿ, namely we compute the alternate sums

X

ε∈{0,1}^l

(−1)^|ε|χ(W_G(ε)∩ {f = 0})

and

X

ε∈{0,1}^l

(−1)^|ε|h

χ(W_G(ε)∩ {f ≥0})−χ(W_G(ε)∩ {f ≤0})i .

Degree formulas for the Euler characteristic of some semialgebraic sets 3

We suppose W = F⁻¹(0) smooth with F = (F₁, . . . , F_k), G = (G₁, . . . , G_l) and k+l ≤n, and we assume that all the intersections between the set W =F⁻¹(0) and the setsG⁻¹_i (0), i= 1, . . . , l are transverse (see Condition(?)on Page 3).

We will see that we can write P

ε(−1)^|ε|χ(W_G(ε)) in terms of the topological degree at innity of the following mappingL dened on Rⁿ×R^k×R^l by:

L(x, λ, µ) = ∇ω(x) +

k

X

i=1

λ_i∇F_i(x) +

l

X

j=1

µ_j∇G_j(x), F(x), µ₁G₁(x), . . . , µ_lG_l(x)

! ,

where ω(x) = ¹₂(x²₁+· · ·+x²_n).

More precisely, we nd the result (c.f. Theorem 2.7):

X

ε∈{0,1}^l

(−1)^|ε|χ(WG(ε)) = (−1)^kdeg_∞L.

To prove this theorem, we will use Morse theory for manifolds with corner, which is an application of stratied Morse theory ([3], [4]). For more details on this theory, we refer to [2].

In my thesis, I also obtained similar results for semialgebraic sets with isolated singu- larities.

The paper is organized as follows: in Section 2, we prove some technical lemmas that relate a Morse index to a topological degree, in order to give a similar result as the one of Szafraniec in [6].Some computations are given in the example at the end of each section. They have been done with a program written by Andrzej Lecki. The author is very grateful to him and Zbigniew Szafraniec for giving her this program.

2. Smooth case In this section, we compute the alternate sums :

X

ε∈{0,1}^l

(−1)^|ε|χ(W_G(ε)),

and as Szafraniece in [6], we want to write it in terms of the topological degree of a polynomial mapping.

Let F = (F₁, . . . , F_k) : Rⁿ → R^k and G = (G₁, . . . , G_l) : Rⁿ → R^l be polynomial maps, with k, l≥1and k+l≤n.

We deneW ={x∈Rⁿ|F(x) = 0}and assume it is a smooth manifold of dimension n−k inRⁿ (or empty). We also make the following transversality assumption:

(?) ∀{i₁, . . . , i_s} ⊆ {1, . . . , l}, ∀x∈W ∩G⁻¹_i

1 (0)∩ · · · ∩G⁻¹_is (0), rank (DF(x), DG_i1(x), . . . , DG_is(x)) =k+s.

We are interested in the topology of the set

W_G(ε) :=W ∩ {x∈Rⁿ|(−1)^ε1G₁(x)≥0} ∩ · · · ∩ {x∈Rⁿ|(−1)^εlG_l(x)≥0}

for ε∈ {0,1}^l and in computing the value of the alternate sum X

ε∈{0,1}^l

(−1)^|ε|χ(W_G(ε)),

where |ε|=

l

X

i=1

ε_i.

Let us remark that for ε ∈ {0,1}^l, W_G(ε) is a manifold with corners, so it is also a Whitney stratied set, whose2^l strata are given by:

S_I^ε_j =

x∈W | (−1)^εiG_i(x)>0if i∈I_j and G_i(x) = 0 if i∈I_j

where for allj ∈ {1, . . . , l},I_j is a subset {i₁, . . . , i_j}of{1, . . . , l}such that i₁ <· · ·<

i_j, and where we writeI_j its complement in{1, . . . , l}. Moreover, we denote byI₀ =∅, and then we haveI₀ ={1, . . . , l}.

For example, when l = 2, G = (G₁, G₂) : Rⁿ → R² and for a given ε = (ε₁, ε₂) ∈ {0,1}², the four strata are:

S_I^ε₀ = {x∈W | G1(x) =G2(x) = 0}, S_I^ε

1 = {x∈W | (−1)^ε1G₁(x)>0, G₂(x) = 0}, S_I^ε0

1 = {x∈W | (−1)^ε2G₂(x)>0, G₁(x) = 0}, S_I^ε

2 = {x∈W | (−1)^ε1G₁(x)>0, (−1)^ε2G₂(x)>0}. We will also consider the following submanifold ofW:

W_Ij =

x∈W | G_i(x) = 0 if i∈I_j, G_i(x)6= 0 if i∈I_j ,

and so we have that for allε∈ {0,1}^l and for all j ∈ {0, . . . , l},S_I^ε_j ⊆W_Ij. We denote byW_Ij the closure of W_Ij in W, that is

W_Ij =

x∈W | G_i(x) = 0 ∀i∈I_j .

Given a polynomial functionρfromRⁿ toR, we are rst going to study the critical points ofρon the dierent strata. We need for this to recall the denition of a correct critical point on a manifold with corners.

Denition 2.1. Let M ∩ {h₁ ≥ 0, . . . , h_k ≥ 0} be a manifold with corners and let us set H := M ∩ {h_i1 = 0, . . . , h_is = 0} ∩ {h_is+1 > 0, . . . , h_ik > 0}. We say that a critical point p of ρ_|H is correct if p is not a critical point of ρ on M ∩ ∪i∈I{h_i = 0}

for any proper subset I of {i₁, . . . , i_s}. Moreover, we say that p is pointing inward (resp. outward) the set M ∩ {h₁ ≥ 0, . . . , h_k ≥ 0} if α_i1 > 0, . . . , α_is > 0 where

∇ρ(p) =

s

X

j=1

α_ij∇h_ij(p) (resp. α_i1 <0, . . . , α_is <0).

We remark that in our case,p∈S_I^ε_j is a correct critical point ofρ|W_G(ε) if and only if pis a critical point ofρ|_W

Ij and is not a critical point ofρ_|W

I for any setI ⊆ {1, . . . , l}

such thatI_j $I.

Degree formulas for the Euler characteristic of some semialgebraic sets 5

For(x, λ, µ)∈Rⁿ×R^k×R^l, we consider the mapping:

L(x, λ, µ) = ∇ρ(x) +

k

X

i=1

λ_i∇F_i(x) +

l

X

j=1

µ_j∇G_j(x), F(x), µ₁G₁(x), . . . , µ_lG_l(x)

! ,

and for(x, λ, µ)∈Rⁿ×R^k×R^l−j,j ∈ {0, . . . , l}, and I_j = (i₁, . . . , il−j), we introduce the 2^l following mappings:

L_I

j(x, λ, µ) = ∇ρ(x) +

k

X

i=1

λ_i∇F_i(x) +

l−j

X

s=1

µ_is∇G_is(x), F(x), µ_i1G_i1(x), . . . , µ_il−jG_il−j(x)

! .

Lemma 2.2. Let p ∈ S_I^ε

j. The point p is a critical point of ρ|_WG(ε) if, and only if, there exists a unique (λ, µ)∈Rⁿ×R^l such that L(p, λ, µ) = 0.

Moreover,

p is correct ⇐⇒µ_i 6= 0 ∀i∈I_j and µ_i = 0 otherwise. Proof. From [6] Lemma 1.2, we know that

pis a critical point of ρ|_S^ε

Ij ⇐⇒ ∃!(λ,µ)¯ ∈R^k×R^l−j, L_I

j(p, λ,µ) = 0.¯ Hence

L(p, λ, µ) = 0 where µ= (µ₁, . . . , µ_l)with µ_i =

0 if i∈I_j

¯

µ_i if i∈I_j . The converse is obvious. It is straightforward to prove the unicity of(λ, µ) .

Now let us suppose that p is correct. As p ∈ S_I^ε_j and L(p, λ, µ) = 0, we necessarily haveµ_i = 0 for all i∈I_j and as p is correct, µ_i 6= 0 for all i∈I_j.

Conversely, ifµ_i = 0 for all i∈I_j and µ_i 6= 0 for all i∈I_j, then

∇ρ(p) +X

λ_i∇F(p) +X

i∈Ij

µ_i∇G_i(p) = 0.

We see thatp is a correct critical point of ρ|_WG(ε). Lemma 2.3. Let p ∈ S_I^ε_j be a correct critical point of ρ|_WG(ε) and (λ, µ) ∈ R^k×R^l such that L(p, λ, µ) = 0. Then p is a correct critical point of ρ|_WG(ε) pointing inward if, and only if, (−1)^εiµ_i <0 for all i∈I_j.

Proof. From the previous lemma, p is a correct critical point of ρ|_WG(ε) if and only if

∃!(λ, µ) such that L(p, λ, µ) = 0 with µ_i = 0 for all i ∈ I_j and µ_i 6= 0 for all i ∈ I_j. Denoting by Π the orthogonal projection on T_pW, we have

Π(∇ρ(p)) = −

l

X

i=1

µ_iΠ(∇G_i(p)) =−X

i∈I_j

µ_iΠ(∇G_i(p)).

We know that for all i∈I_j, Π((−1)^εi∇G_i(p)) points inward W_G(ε), so Π(∇ρ(p)) = X

i∈Ij

(−(−1)^εiµi)Π((−1)^εi∇Gi(p))

points inward if and only if −(−1)^εiµ_i >0, i.e. (−1)^εiµ_i <0for all i∈I_j. In the next lemma, we want to relate the sign of the dierential of L at one zero (p, λ, µ) of Lto the Morse index at p of ρ restricted to the stratum that contains p. Lemma 2.4. Let p ∈ S_I^ε_j be a correct critical point of ρ|_WG(ε) and let (λ, µ) ∈ R^k× R^l be such that L(p, λ, µ) = 0. Then p is a Morse critical point if, and only if, det(DL(p, λ, µ))6= 0. Moreover, we get that

sign det(DL(p, λ, µ)) =



 Y

i∈Ij

sign(G_i(p))Y

i∈I_j

sign(µ_i)



(−1)^{τ(p)+k+l−j},

where τ(p) is the Morse index ofρ|_S^ε

Ij at p. Proof. Let (p, λ, µ) be a zero ofL:

DL(p, λ, µ) =





























(?) B ∇G₁(p). . .∇G_l(p)

tB (0) (0)

µ^t₁∇G₁(p) ...

µ^t_l∇G_l(p)

(0)

G₁(p) ...

G_l(p)



























 ,

with B = (∇F₁(p), . . . ,∇F_k(p))∈M_n,k(R)and (?)∈M_n(R).

We know thatp∈S_I^ε_j soµ_i = 0 ∀i∈I_j and G_i(p) = 0 ∀i∈I_j, which we denote by I_j = (i₁, . . . , il−j). Then we get:

DL(p, λ, µ) =



 Y

i∈Ij

G_i(p)Y

i∈Ij

µ_i

































(?) B ∇G_i1(p). . .∇G_il−j(p)

tB (0) (0)

t∇G_i1(p) ...

t∇G_il−j(p)

(0) (0)



























 ,

=



 Y

i∈I_j

Gi(p)Y

i∈Ij

µi



DL_Ij(p, λ,µ),¯

Degree formulas for the Euler characteristic of some semialgebraic sets 7

where µ¯∈R^l−j is such that µ¯_i =µ_i ∀i∈I_j. From [6] Lemma 1.4, we know that

pis Morse ⇐⇒det(DL_I

j(p, λ,µ))¯ 6= 0, and we have

det (DL(p, λ, µ)) =



 Y

i∈I_j

G_i(p)Y

i∈Ij

µ_i



det

DL_Ij(p, λ,µ)¯ .

So

det(DL(p, λ, µ))6= 0.

Finally, still from [6] Lemma 1.4, we know that sign det

DL_I

j(p, λ,µ)¯

= (−1)^τ(p)(−1)^k+l−j, hence we obtain

sign det (DL(p, λ, µ)) =



 Y

i∈Ij

sign(G_i(p))Y

i∈Ij

sign(µ_i)



(−1)^{τ(p)+k+l−j}.

Remark 2.5. If k +l = n and p is a correct critical point of ρ|S_I^ε

0, its Morse index τ(p) is 0.

Now we chooseρ(x) = ω(x) = ¹₂(x²₁+· · ·+x²_n) so we have∇ω(x) =x. For all positive constantR, we set:

B_R^n+k+l ={(x, λ, µ)∈Rⁿ×R^k×R^l | kxk²+kλk²+kµk² ≤R²}, S_R^n+k+l−1 ={(x, λ, µ)∈Rⁿ×R^k×R^l | kxk²+kλk²+kµk² =R²}.

We would like to computedeg_∞L, the topological degree at innity of L

kLk :S_R^n+k+l−1 → S^n+k+l−1 for some R >0but we need to prove rst that it is well dened.

Lemma 2.6. The set L⁻¹(0) is compact.

Proof. Let us study the vanishing set of L. We easily nd that

L⁻¹(0) =[

j,Ij

n

(p, λ, µ)∈Rⁿ×R^k×R^l | (p, λ,µ)¯ ∈L⁻¹

Ij (0) with

¯

µ_i =µ_i ∀i∈I_j and µ_i = 0 otherwiseo , which we can also write

L⁻¹(0) =[

j,Ij

L⁻¹

Ij (0)×

0 ^j (up to changes of indices in the coordinate system).

From [6] Lemma 2.1, the set L⁻¹

Ij (0) is compact for all j ∈ {0, . . . , l} and so is the productL⁻¹

Ij (0)× {0}. As the union onj andI_j is nite, we can conclude that the set

L⁻¹(0) is compact.

Theorem 2.7. LetR >0be such thatL⁻¹(0) $B_R^n+k+l. If the setsW andG⁻¹_i (0), i = 1, . . . , l satisfy the transversality condition (?), then

X

ε

(−1)^|ε|χ(W_G(ε)) = (−1)^kdeg_∞L.

Proof. As in the proof of Theorem 2.2 in [6], we can choose a function ω˜ : Rⁿ → R equal to ω onRⁿ\B_R, such that ω|˜ _W and ω|˜ _WG(ε), with ε∈ {0,1}^l, have only Morse correct critical points, and nally such thatω˜uniformly approaches ωfor the Whitney C²-topology.

For(x, λ, µ)∈Rⁿ×R^k×R^l, we consider L(x, λ, µ) =˜ ∇˜ω(x) +X

i

λ_i∇F_i(x) +X

i

µ_i∇G_i(x), F(x), µ₁G₁(x), . . . , µ_lG_l(x)

! .

As L⁻¹(0) ⊆ B_R^n+k+l, for ω˜ close enough to ω, we also have that L˜⁻¹(0) ⊆ B_R^n+k+l. Let (x, λ, µ) ∈ L˜⁻¹(0). Then there exists j ∈ {0, . . . , l} such that x ∈ W_Ij and from Lemma 2.2, x is a critical point of ω|˜ _W

Ij. But we have supposed that ω|˜ _WIj only had Morse correct critical points, so from Lemma 2.4, we know that detDL(x, λ, µ)˜ 6= 0. Thus,0∈Rⁿ×R^k×R^l is a regular value of L˜. Moreover, the set L˜⁻¹(0) is nite.

We now want to compute the degree ofL˜, which is equal to the degree ofL since if ω and ω˜ are close enough, _kLk^L and _k^LLk^˜˜ are homotopic on a suciently big sphere.

Let us deneCL˜ ={(p, λ, µ)∈Rⁿ×R^k×R^l | L(p, λ, µ) = 0}˜ the zero set ofL˜. Then we consider:

F_Ij ={µ∈R^l | µ_i = 0 ∀i∈I_j, µ_i 6= 0 ∀i∈I_j} and the following subset of CL˜:

C_I^ε

j =

(p, λ, µ)∈CL˜ | µ∈F_Ij, (−1)^εiG_i(p)>0 if i∈I_j, (−1)^εi+1µ_i >0 if i∈I_j . From Morse theory for manifolds with corners ([2]), for a givenε, we have

χ(WG(ε)) =

l

X

j=0

X

Ij

X

p∈Πn(C_Ij^ε)

(−1)^τ(p),

whereτ(p) is the Morse index of ω|˜ _WIj atp andΠ_n the projection ofRⁿ×R^k×R^l on Rⁿ. Hence

X

ε

(−1)^|ε|χ(W_G(ε)) =X

ε

(−1)^|ε|

l

X

j=0

X

Ij

X

p∈Πn(C^ε

Ij)

(−1)^τ(p).

Degree formulas for the Euler characteristic of some semialgebraic sets 9

But for(p, λ, µ)∈C_I^ε

j, we know that signDL(p, λ, µ) = sign˜



 Y

i∈Ij

G_i(p)



sign



 Y

i∈I_j

µ_i



(−1)^{k+l−j+τ(p)}

and we easily see that sign



 Y

i∈Ij

G_i(p)



sign



 Y

i∈I_j

µ_i



=Y

i∈Ij

(−1)^εiY

i∈I_j

(−1)^εi+1

=

l

Y

i=1

(−1)^εi

!

(−1)^card(Ij)= (−1)^|ε|(−1)^l−j.

As

deg_∞L˜ =X

ε l

X

j=0

X

Ij

X

p∈Πn(C_Ij^ε )

sign



 Y

i∈I_j

Gi(p)



sign



 Y

i∈Ij

µi



(−1)^τ(p)(−1)^k+l−j,

and

deg_∞L˜ = deg_∞L, we can conclude that

X

ε

(−1)^|ε|χ(WG(ε)) = (−1)^kdeg_∞L.

Example 2.8.

f(x₁, x₂) = −(x₁+ 3)³x₂−3x²₁+x²₂+x₁, G(x₁, x₂) = x²₁+x²₂−30.

Here, the transversality condition (?) is satised.

We have

L:R²⁺¹⁺¹ −→ R²⁺¹⁺¹

(x, λ, µ) 7−→ (∇ω(x) +λ∇f(x) +µ∇G(x), f(x), µG(x)).

We get that deg_∞L=−2 and thus X

ε∈{0,1}

(−1)^|ε|χ(W_G(ε)) = χ(f⁻¹(0)∩ {G≥0})−χ(f⁻¹(0)∩ {G≤0}) =−deg_∞L= 2.

The setsf⁻¹(0)∩ {G≥0} andf⁻¹(0)∩ {G≤0} are respectively four and two lines so are respectively homotopic to four and two points. Hence their respective Euler characteristic are 4 and 2. Thus,

χ(f⁻¹(0)∩ {G≥0})−χ(f⁻¹(0)∩ {G≤0}) = 4−2 = 2.

Figure 1. Curves G⁻¹(0) and f⁻¹(0). References

[1] J. W. Bruce. Euler characteristics of real varieties. Bull. London Math. soc., 22:547552, 1990.

[2] N. Dutertre. On topological invariants associated with a polynomial with isolated critical points.

Glasgow Math. J., 46:323334, 2004.

[3] M. Goresky and R. Mac Pherson. Stratied Morse Theory, volume 89. Springer-Verlag, 1988.

[4] H.A. Hamm. On stratied Morse theory. Topology, 38:427438, 1999.

[5] Z. Szafraniec. On the Euler characteristic of analytic and algebraic sets. Topology, 25:411414, 1986.

[6] Z. Szafraniec. The Euler characteristic of algebraic complete intersections. J. reine angew. Math., 397:194201, 1989.

Aix-Marseille Université, CNRS, Centre Gilles-Gaston Granger, Aix-en-Provence, France

E-mail address: julie.lapebie@univ-amu.fr