The total Betti number of the intersection of three real quadrics
aa r X i v : . [ m a t h . A T ] N ov THE TOTAL BETTI NUMBER OF THE INTERSECTION OFTHREE REAL QUADRICS
A. LERARIO
Abstract.
We prove the bound b ( X ) ≤ n ( n +1) for the total Betti number ofthe intersection X of three quadrics in R P n . This bound improves the classicalBarvinok’s one which is at least of order three in n . Introduction
In this paper we address the problem of bounding the total Betti number ofthe intersection X of three real quadrics in R P n . In the case X were a smooth,complete intersection , then its total Betti number can be easily bounded usingSmith’s theory: its equations can be real perturbed as not to change its topology(it is smooth) and to make its complex points also a smooth complete intersection;in the case of a complete intersection X C of three quadrics in C P n it is possible tocompute its Betti numbers using Hirzebruch’s formula and this would give a boundof the type b ( X ) ≤ b ( X C ) ≤ p ( n ) where p is a polynomial of degree two.In the general case, i.e. when we make no regularity assumption on X , the problemturns out to be more complicated. If we simply naively perturb the equationsdefining X we can of course make its complex point to be smooth, but then thetopology of the real part would have change.The very first attempt to bound the topology of the intersection of k quadrics in R P n is to use the well known Oleinik-Petrovskii-Thom-Milnor inequality (see [6]),which gives the estimate : b ( X ) ≤ O ( k ) n . Surprisingly enough it turns out that the fact that X is defined by quadratic equa-tions allows to interchange the role of the two numbers n and k and to get theclassical Barvinok’s bound (see [4]): b ( X ) ≤ n O ( k ) . The hidden constant in the exponent for this estimate is at least two, as noticedalso by the authors of [5], where a more refined estimate is presented (but of thesame leading order).In particular in the case X is the intersection of three quadrics in R P n this classicalestimates would give b ( X ) ≤ n O (3) . The passage from Oleinik-Petrovskii-Thom-Milnor bound to Barvinok’s one is es-sentially made using a kind of duality argument, which works for the quadratic
SISSA, Trieste. According to [6] in this context the notation f ( n ) = O ( n ) means that there exists a naturalnumber b such that the inequality f ( n ) ≤ bn hods for every n ∈ N . case, between the number of variables and the number of equations. This ideaappeared for the first time in the paper [2] and we explain it now. If we have thequadratic forms q , . . . , q k on R n +1 , then we can consider their linear span L in thespace of all quadratic forms. The arrangement of L with respect to the subset Z ofdegenerate quadratic forms (those with at least one dimensional kernel) determinesin a very precise way the topology of the base locus: X = \ q ∈ L \{ } { [ x ] ∈ R P n | q ( x ) ≤ } . The simplest invariant we can associate to a quadratic form q is its positive inertiaindex i + ( q ), namely the maximal dimension of a subspace V ⊂ R n +1 such that q | V is positive definite. In a similar fashion we are led to consider for j ∈ N the sets:Ω j = { q ∈ L \{ } | i + ( q ) ≥ j } . The spirit of the mentioned duality is in this procedure of replacing the originalframework with the filtration:Ω n +1 ⊆ Ω n ⊆ · · · ⊆ Ω ⊆ Ω . This duality is widely investigated in the paper [3], where a spectral sequenceconverging to the homology of X is studied; this spectral sequence has second term E i,j isomorphic to H n − i ( L, Ω j +1 ) . Thus, at a first approximation, the previouscohomology groups can be taken as the homology of X and as long as we considerfiner properties of the arrangement of L , then new information on the topology ofits base locus is obtained. It is remarkable that only using this approximation theclassical Barvinok’s bounds can be recovered; in this setting they can be formulatedin the form:(1) b ( X ) ≤ n + 1 + X j ≥ b (Ω j +1 )The introduction of the full methods from [3] made also possible in some cases tostrongly improve the classical bounds. For example in the paper [11] the authorproves that the total Betti number of the intersection X of two quadrics in R P n is bounded by 3 n + 2; in the same papers are provided also bounds linear in k foreach specific Betti number of X. The intersection of L with the set of degenerate forms Z is customary called thespectral variety C of the base locus X . In this paper we present the idea thatin the case X is the intersection of three quadrics, then its topological complexityis essentially that of its spectral variety C ; this variety is in fact the “difference”between the various sets Ω j and thus the sum in the right hand side of (1) in acertain sense is “bounded” by b ( C ) . This idea originally appears in [1] in the regularcase. Here enters the deep connection, generally called Dixon’s correspondence,between the intersection of three real quadrics in R P n and curves of degree n + 1on R P : in the case X is a smooth, complete intersection the corresponding curveis the projectivization of the spectral variety (see [9] and [8]). In our framework X is no longer smooth, nor a complete intersection, and a pertubative approach isintroduced to study it; this approach associates to X a smooth curve in S whichreplaces the role of the spectral variety (indeed in the regular case this curve isthe double cover of the curve given by Dixon’s correspondence). We relate thecomplexity of X to that of this new curve and using a Harnack’s type argument onthe sphere will give us the mentioned bound b ( X ) ≤ n ( n + 1) . HE TOTAL BETTI NUMBER OF THE INTERSECTION OF THREE REAL QUADRICS 3
Acknowledgements
The author is grateful to his teacher A. A. Agrachev: the main idea of this paperis a natural development of his intuition.2.
General settings
We recall in this section a general construction to study the topology of intersec-tion of real quadrics. We set Q ( n + 1) for the space of real quadratic forms in n + 1variables; if q , . . . , q k belong to this space, then we can consider their common zerolocus X in R P n : X = V R P n ( q , . . . , q k ) , q , . . . , q k ∈ Q ( n + 1)To study the topology of X we introduce the following auxiliary construction. Wedenote by q the ( k + 1)-ple ( q , . . . , q k ) and consider the map q : S k → Q ( n + 1),defined by ω ωq = ω q + · · · + ω k q k , ω = ( ω , . . . , ω k ) ∈ S k . This map places the unit sphere S k linearly into the space of quadratic forms, inthe direction of the chosen quadrics. For a given quadratic form p ∈ Q ( n + 1)we denote by i + ( p ) its positive inertia index, namely the maximal dimension ofa subspace of R n +1 such that the restriction of p to it is positive definite. For a family of quadratic forms depending on some parameters, like the map q describes,we consider the geometry of this function on the parameter space. We are thusnaturally led to define the sets:Ω j = { ω ∈ S k | i + ( ωq ) ≥ j } , j ∈ N The following theorem relates the topological complexity of X to that of the setsΩ j . For a semialgebraic set S we define b ( S ) to be the sum of its Betti numbers . Theorem 1 (Topological complexity formula) . b ( X ) ≤ n + 1 + X j ≥ b (Ω j +1 ) Proof.
Consider the topological space B = { ( ω, [ x ]) ∈ S k × R P n | ( ωq )( x ) > } together with its two projections p : B → S k and p : B → R P n . The imageof p is easily seen to be R P n \ X and the fibers of this map are contractible sets,hence p gives a homotopy equivalence B ∼ R P n \ X. Consider now the projection p ; for a point ω ∈ S the fiber p − ( ω ) has the homotopy type of a projectivespace of dimension i + ( ωq ) −
1, thus the Leray spectral sequence for p convergesto H ∗ ( R P n \ X ) and has the second terms E i,j isomorphic to H i (Ω j +1 ) . A detailedproof of the previous statements can be found in [3]. Since rk( E ∞ ) ≤ rk( E )then b ( R P n \ X ) ≤ P j ≥ b (Ω j +1 ). Recalling that by Alexander-Pontryagin duality H n −∗ ( X ) ≃ H ∗ ( R P n , R P n \ X ) , then the exactness of the long cohomology exactsequence of the pair ( R P n , R P n \ X ) gives the desired inequality. (cid:3) From now on every homology group is assumed with Z coefficients, unless differently stated;the same remark applies to Betti numbers. A. LERARIO
Remark . A more refined formula for b ( X ) follows by considering a different spec-tral sequence directly converging to H n −∗ ( X ). In fact by [3] there exists such aspectral sequence ( E r , d r ) r ≥ with second term E i,j ≃ H i ( B, Ω j +1 ), where B is theunit ball in R k +1 and Ω j +1 ⊆ ∂B. If we let µ be the maximum of i + on S k and ν beits minumum, then we get b ( X ) ≤ rk( E ) ≤ n +1 − µ − ν )+ P ν +1 ≤ j +1 ≤ µ b (Ω j +1 ).The paper [3] contains a description of the second differential of this spectral se-quence, which happens to be related with the set of points on S k where ωq hasmultiple eigenvalues, together with applications. Remark . By universal coefficients theorem, the previous bound is valid also forthe total Betti number of X with coefficients in Z (but on the right hand side Z coefficients are still assumed).The previous formula, together with some results from [6], can be used to givethe classical Barvinok’s estimate (see the paper [4]). Corollary 2 (Barvinok’s estimate) . b ( X ) ≤ ( n + 1) O (2 k +2) . Proof.
Let us fix a scalar product; then the rule h x, ( ωQ ) x i = q ( x ) defines a sym-metric matrix ωQ whose number of positive eigenvalues equals i + ( ωq ) . Considerthe polynomial det( ωQ − tI ) = a ( ω ) + · · · + a n ( ω ) t n ± t n +1 ; then by Descartes’rule of signs the positive inertia index of ωQ is given by the sign variation in thesequence ( a ( ω ) , . . . , a n ( ω )) . Thus the sets Ω j +1 are defined on the sphere S k bysign conditions (quantifier-free formulas) whose atoms are polynomials in k + 1variables and of degree less than n + 1. For such sets we have the estimate, provedin [6]: b (Ω j +1 ) ≤ ( n + 1) O (2 k +1) . Putting all them together we get: b ( X ) ≤ n + 1 + n X j =0 b (Ω j +1 ) ≤ ( n + 1) O (2( k +1)) (notice that k + 1 is the number of quadrics cutting X ). (cid:3) Remark . The paper [4] contains the bound for the set S of solutions of k quadraticinequalities in R n +1 , which is b ( S ) ≤ ( n +1) O ( k ) . The set X in R P n can be viewed asdouble covered by a subset X ′ in R n +1 defined by k +2 quadratic inequalities (thosedefining X together with the quadratic equation for the unit sphere); by the transferexact sequence b ( X ) ≤ b ( X ′ ) and by Barvinok’s estimate b ( X ′ ) ≤ ( n + 1) O ( k +2) ;since the constant hidden in the previous exponent is at least two, then we get thesame order as in the previous corollary.A more refined bound in the general case can be found in [5].3. Preliminaries on perturbations
Let now p ∈ Q ( n + 1) be a positive definite form (we will use the notation p > ǫ > k ∈ N let us define the sets:Ω n − j ( ǫ ) = { ω ∈ S k | i − ( ωq − ǫp ) ≤ n − j } where i − denotes the negative inertia index, i.e. i − ( ωq − ǫp ) = i + ( ǫp − ωq ) . Thefollowing lemma relates the topology of Ω j +1 and of its perturbation Ω n − j ( ǫ ) . HE TOTAL BETTI NUMBER OF THE INTERSECTION OF THREE REAL QUADRICS 5
Lemma 3.
For every positive definite form p ∈ Q ( n + 1) and for every ǫ > sufficiently small b (Ω j +1 ) = b (Ω n − j ( ǫ )) . Proof.
Let us first prove that Ω j +1 = S ǫ> Ω n − j ( ǫ ) . Let ω ∈ S ǫ> Ω n − j ( ǫ ); then there exists ǫ such that ω ∈ Ω n − j ( ǫ ) for every ǫ < ǫ. Since for ǫ small enoughi − ( ωq − ǫp ) = i − ( ωq ) + dim(ker( ωq ))then it follows thati + ( ωq ) = n + 1 − i − ( ωq ) − dim(ker ωq ) ≥ j + 1 . Viceversa if ω ∈ Ω j +1 the previous inequality proves ω ∈ Ω n − j ( ǫ ) for ǫ small enough,i.e. ω ∈ S ǫ> Ω n − j ( ǫ ) . Notice now that if ω ∈ Ω n − j ( ǫ ) then, eventually choosing a smaller ǫ , we mayassume ǫ properly separates the spectrum of ω and thus, by continuity of the map q , there exists U open neighborhood of ω such that ǫ properly separates also thespectrum of ηq for every η ∈ U (see [10] for a detailed discussion of the regularity ofthe eigenvalues of a family of symmetric matrices). Hence every η ∈ U also belongsto Ω n − j ( ǫ ). From this consideration it easily follows that each compact set in Ω j +1 is contained in some Ω n − j ( ǫ ) and thuslim −→ ǫ { H ∗ (Ω n − j ( ǫ )) } = H ∗ (Ω j +1 ) . It remains to prove that the topology of Ω n − j ( ǫ ) is definitely stable in ǫ going to zero.Consider the semialgebraic compact set S n − j = { ( ω, ǫ ) ∈ S k × [0 , ∞ ) | i − ( ωq − ǫp ) ≤ n − j } . By Hardt’s triviality theorem (see [7]) we have that the projection ( ω, ǫ ) ω is a locally trivial fibration over (0 , ǫ ) for ǫ small enough; from this the conclusionfollows. (cid:3) Let us now move to specific properties of pencils of three quadrics.We recall that the space of degenerate forms Z = { p ∈ Q ( n +1) | ker( p ) = 0 } admitsa semialgebraic Nash stratification Z = ` Z i such that its singularities belong tostrata of codimension at least three in Q ( n + 1); this is a classical result and thereader can see [2] for a direct proof. Lemma 4.
There exists a positive definite form p ∈ Q ( n + 1) such that for every ǫ > small enough the curve C ( ǫ ) = { ω ∈ S | ker ( ωq − ǫp ) = 0 } is a smooth curve in S such that the difference of the index function ω i − ( ωq − ǫp ) on adjacent components of S \ C ( ǫ ) is ± . Proof.
Let Q + be set of positive definite quadratic forms in Q ( n + 1) and considerthe map F : S × Q + defined by ( ω, p ) ωq − p. Since Q + is open in Q , then F is a submersion and F − ( Z ) is Nash-stratifiedby ` F − ( Z i ) . Then for p ∈ Q + the evaluation map ω f ( ω ) − p is transver-sal to all strata of Z if and only if p is a regular value for the restriction ofthe second factor projection π : S × Q + → Q + to each stratum of F − ( Z ) = ` F − ( Z i ) . Thus let π i = π | F − ( Z i ) : F − ( Z i ) → Q + ; since all datas are smooth A. LERARIO semialgebraic, then by semialgebraic Sard’s Lemma (see [7]), the set Σ i = { ˆ q ∈Q + | ˆ q is a critical value of π i } is a semialgebraic subset of Q + of dimension strictlyless than dim( Q + ) . Hence Σ = ∪ i Σ i also is a semialgebraic subset of Q + of di-mension dim(Σ) < dim( Q + ) and for every p ∈ Q + \ Σ the map ω f ( ω ) − p is transversal to each Z i . Since Σ is semialgebraic of codimension at least one,then there exists p ∈ Q + \ Σ such that { tp } t> intersects Σ in a finite number ofpoints, i.e. for every ǫ > ǫp ∈ Q + \ Σ . Since the codimensionof the singularities of Z are at least three, then for p ∈ Q + \ Σ and ǫ > { ω ∈ S | ker( ωq − ǫp ) = 0 } is smooth. Moreover if z is a smoothpoint of Z , then its normal bundle at z coincides with the sets of quadratic forms { λ ( x ⊗ x ) | x ∈ ker( z ) } λ ∈ R then also the second part of the statement follows. (cid:3) Essentially lemma 4 tells that we can perturb the map ω ωq in such a waythat the set of points where the index function can change is a smooth curve on S ;lemma 3 tells us how to control the topology of the sets Ω j +1 after this perturbation. Remark . In higher dimension all that we can do is perturb the map q as to makeit as best as possible, i.e. transversal to all strata of Z = ` Z i ; for example in thecase of four quadrics we can make the hypersurface { ω ∈ S | ker( ωq − ǫp ) = 0 } areal algebraic manifold of dimension two with at most isolated singularities.4. Harnack’s type inequalities
In this section we bound the topology of a smooth curve C ⊂ S (as the aboveone) in a way similar to Harnack’s classical bounds for smooth curves in R P . Westart with the following lemma.
Lemma 5.
Let G ∈ R [ ω , . . . , ω ] be a homogeneous polynomial. The set Σ of allhomogeneous polynomials F of a fixed degree d such that V C P ( F, G ) is not a smoothcomplete intersection in C P is a proper real algebraic set in R [ ω , . . . , ω ] ( d ) .Proof. The set Σ C of homegeneous polynomials H with complex coefficients anddegree d such that V C P ( H, G ) is not a complete intersection in C P is clearly aproper complex algebraic subset of C [ ω , . . . , ω ] ( d ) ≃ C N , where N = (cid:0) dd (cid:1) . LetΣ C be defined by the vanishing of certain polynomials f , . . . , f l in C [ z , . . . , z N ].Since Σ equals Σ C ∩ R N , then it is real algebraic; it remains to show it is proper .Suppose not. Then Σ C ∩ R N = R N , which means that each of the f i vanishesidentically over the reals. In particular, fixing all but one variables in f i we havea complex polynomial in one variable which has infinite zeroes, hence it must bezero. Iterating the reasoning for each variable this would give that each f i is zero,which is absurd since Σ C was a proper algebraic set. (cid:3) Lemma 6.
Let f ∈ R [ ω , ω , ω ] be a polynomial of degree d such that C = { ( ω , ω , ω ) ∈ S | f ( ω , ω , ω ) = 0 } is a smooth curve; then the number of its ovals is at most d ( d −
2) + 2 .Proof. If f is homogeneous, then C is the double cover of a smooth curve C ′ in R P of degree d ; hence by Harnack’s inequality b ( C ′ ) ≤ ( d − d −
2) + 2 . Bythe transfer exact sequence b ( C ) ≤ b ( C ′ ), which in this case gives the bound( d − d −
2) + 2 ≤ d ( d −
2) + 2 for the number of the ovals of C .Assume now f is not homogeneous and let F ∈ R [ ω , . . . , ω ] be its homogenization HE TOTAL BETTI NUMBER OF THE INTERSECTION OF THREE REAL QUADRICS 7 (the new variable is ω ); let also G be the polynomial G ( ω , . . . , ω ) = ω + ω + ω − ω . Using this setting we have that the curve C coincides with: V R P ( F, G ) ⊂ R P (there are no solutions on the hyperplane { ω = 0 } to F = G = 0). By lemma 5there exists a real perturbation F ǫ of the polynomial F , homogeneous and of thesame degree of F , such that V C P ( F ǫ , G ) ⊂ C P is a smooth complete intersection in C P . Moreover since the perturbation wasreal, then by Smith’s theory the total Betti number of V R P ( F ǫ , G ) is bounded bythat of V C P ( F ǫ , G ); on the other hand since V R P ( F, G ) was smooth, then a smallperturbation of its equations does not change its topology, hence the total Bettinumber of C = V R P ( F, G ) also is bounded by that of V C P ( F ǫ , G ) . It remains toprove that for the complete intersection V C P ( F ǫ , G ) the bound on its topologicalcomplexity is 2 d ( d −
2) + 4. To this end notice that Y = V C P ( F ǫ , G ) is a smoothcomplex curve of degree 2 d ; hence if we let K Y be its canonical bundle the adjunc-tion formula reads K Y = O C P ( d − | Y = ( O C P (1) | Y ) ⊗ ( d − . Since the degree of K Y is 2 g ( Y ) −
2, then2 g ( Y ) − d − O C P (1) | Y ) = ( d − d. Since b ( Y ) = 2 g ( Y ) + 2 this concludes the proof. (cid:3) Corollary 7.
There exists a positive definite form p in Q ( n +1) such that for every ǫ > small enough the smooth curve C ( ǫ ) = { ω ∈ S | ker ( ωq − ǫp ) = 0 } has at most ( n + 1)( n −
1) + 2 ovals.Proof.
Let p be given by lemma 4. Fix a scalar product in such a way that eachquadratic form can be identified with a real symmetric matrix, as in the proofof Barvinok’s estimate. Thus ωQ and P are the symmetric matrices associatedrespectively to ωq and p . The conclusion follows simply by applying the previouslemma to the polynomial f ( ω , ω , ω ) = det( ωQ − ǫP ), which has degree n + 1. (cid:3) We recall in this section also the following elementary fact.
Lemma 8.
Let Ω ⊂ S be a surface with boundary ∂ Ω = ∅ . Then: b (Ω) = b ( ∂ Ω) Proof.
By additivity of the formula it is sufficient to prove it in the case Ω isconnected. In this case Ω is homotopy equivalent to the sphere S minus b ( ∂ Ω)points and thus b (Ω) = 1 and b (Ω) = b (Ω) − . (cid:3) The total Betti number of the intersection of three realquadrics
The aim of this section is to prove the following theorem, which estimates thetotal Betti number of the intersection X of three real quadrics in R P n . Notice thatwe do not require any nondegeneracy assumption.
Theorem 9.
Let X be the intersection of three quadrics in R P n . Then: b ( X ) ≤ n ( n + 1) A. LERARIO
Proof.
We use the refined formula b ( X ) ≤ n + 1 − µ − ν ) + P ν +1 ≤ j +1 ≤ µ b (Ω j +1 )for the total Betti number of X ; if we use only the topological complexity for-mula the estimate we can produce is a bit worst, but always of the type n +terms of lower degree). By lemma 4 there exists a positive definite form p suchthat for every ǫ > C ( ǫ ) = { ω ∈ S | ker( ωq − ǫp ) = 0 } is smooth; moreover by lemma 3 for such a p and for ǫ > b (Ω j +1 ) = b (Ω n − j ( ǫ )). This in particular gives b ( X ) ≤ n + 1 − µ − ν ) + X ν +1 ≤ j +1 ≤ µ b (Ω n − j ( ǫ )) . Since for each ν + 1 ≤ j + 1 ≤ µ the set Ω n − j ( ǫ ) is a submanifold of S withnonempty boundary, then by lemma 8: b (Ω n − j ( ǫ )) = b ( ∂ Ω n − j ( ǫ )) . In particular P b (Ω n − j ( ǫ )) equals P b ( ∂ Ω n − j ( ǫ )) , where in both cases the sum ismade over the indexes ν + 1 ≤ j + 1 ≤ µ . The second part of lemma 4 impliesnow that each of the ovals of C ( ǫ ) belongs to the boundary of exactly one of theΩ n − j ( ǫ ) , ν + 1 ≤ j + 1 ≤ µ. This implies that the previous sum P b ( ∂ Ω n − j ( ǫ ))equals exactly the number c of ovals of C ( ǫ ) . In particular this gives: b ( X ) ≤ n + 1 − µ − ν ) + c. If µ = ν, then b ( X ) ≤ n + 1; thus we may assume 2( µ − ν ) ≥
2. Corollary 7 tellsthat c ≤ ( n + 1)( n −
1) + 2, which finally gives: b ( X ) ≤ n + 1 − n + 1)( n −
1) + 2 = n ( n + 1) . (cid:3) Remark . Since in the previous proof the sets Ω n − j ( ǫ ) and their boundaries weresemialgebraic subsets of S , then their Betti numbers with coefficients in Z coincidewith those with coefficient in Z ; moreover by the universal coefficient theorem b ( X ; Z ) ≤ b ( X ) and thus we also have: b ( X ; Z ) ≤ n ( n + 1) . Remark . If we define the map q : R n +1 → R whose components are ( q , q , q ),then the intersection of the three quadrics defined by the vanishing of q , q and q equals { [ x ] ∈ R P n | q ( x ) = 0 } . In a similar way if K ⊂ R is a closed polyhedralcone, we may define (by slightly abusing of notations) the set q − ( K ) = { [ x ] ∈ R P n | q ( x ) ∈ K } . Such a set is the set of the solutions of a system of three quadraticinequalities in R P n and using the spectral sequence of [3] and a similar argument asabove it is possible to prove a bound quadratic in n for its topological complexity.We leave the details to the reader. Remark . In the case X is a complete intersection of quadrics in R P n , estimateson the number of its connected components are given in [8]. In particular, followingthe notations of [8], we can denote by B kr ( n ) the maximum value that the k -th Bettinumber of an intersection (not necessarily complete) of r + 1 quadrics in R P n canhave. There it is proved that for complete intersections B ( n ) ≤ l ( l −
1) + 2 , l = [ n/
2] + 1 . HE TOTAL BETTI NUMBER OF THE INTERSECTION OF THREE REAL QUADRICS 9
The reader should notice that, in accordance with our result, the estimate is qua-dratic in n ; our bound tells in particular that this quadratic estimate holds for every Betti number and also without regularity assumptions.
References [1] A. A. Agrachev:
Homology of intersections of real quadrics , Soviet Math. Dokl., 1988, v.37,493–496 (translated in Soviet Math. Dokl. 37, 1988, no. 2).[2] A. A. Agrachev:
Topology of quadratic maps and hessians of smooth maps , Itogi nauki.VINITI. Algebra. Topologiya. Geometriya, 1988, v.26, 85–124.[3] A. A. Agrachev, A. Lerario:
Systems of quadratic inequalities , arXiv:1012.5731v2.[4] A. I. Barvinok:
On the Betti numbers of semialgebraic sets defined by few quadratic inequal-ities , Discrete and Computational Geometry , 22:1-18 (1999).[5] S. Basu, M. Kettner:
A sharper estimate on the Betti numbers of sets defined by quadraticinequalities , Discrete Comput. Geom., 2008, v.39, 734–746[6] S. Basu, R. Pollack, M-F. Roy:
Algorithms in Real Algebraic Geometry , Springer.[7] J. Bochnak, M. Coste, M-F. Roy:
Real Algebraic Geometry , Springer-Verlag, 1998.[8] A. Degtyarev, I. Itenberg, V. Kharlamov:
On the number of connected components of acomplete intersection of real quadrics , arXiv:0806.4077[9] A. C. Dixon,
Note on the Reduction of a Ternary Quantic to Symmetrical Determinant ,Proc. Cambridge Philos. Soc. 2 (1900–1902), 350–351.[10] T. Kato:
Perturbation theory for linear operators , Springer, 1995.[11] A. Lerario,