The total Betti number of the intersection of three real quadrics

Lerario, Antonio

doi:10.1515/advgeom-2014-0013

Cited by 6 publications

(9 citation statements)

References 17 publications

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“…i − (ωq − ǫp) = i + (ǫp − ωq). In particular the Betti numbers of Ω j+1 and of its perturbation Ω n−j (ǫ) are the same, as proved in the following lemma from [16].…”

Section: Transversality Argumentssupporting

confidence: 52%

“…Recall that equation ( 19) was for a complete intersection of multidegree (2, δ) in CP 3 ; specifying it to this case δ = n + 1 we get b(C) ≤ 2n 2 + O(n), which plugged into the previous inequality gives: b(X) ≤ n 2 + O(n). (indeed Theorem 1 of [16] gives the refined bound b(X) ≤ n 2 + n). We notice now that in this case: B(3, n) = n 2 + O(n).…”

Section: Examplesmentioning

confidence: 98%

“…The following is a variation of Lemma 4 of [16] and describes the structure of the sets of degenerate quadratic forms on the 'perturbed sphere'. We recall that the space Z of all degenerate forms in Q n+1 admits the semialgebraic Nash stratification Z = N r where N r = Z (r) \Z (r+1) (as above we use the linear identification between quadratic forms and symmetric matrices).…”

Section: Transversality Argumentsmentioning

confidence: 99%

“…Equivalently we can consider their homogenization g 1 = h F − ǫω 2d 0 = 0 and g 2 = ω 2 − ω 2 0 = 0 and their common zero locus in RP k : since there are no common solutions on {ω 0 = 0} (the hyperplane at infinity) these two equations still define {F | S k−1 = ǫ}. By Fact 1 in [16] it follows that we can real perturb the coefficients of g 1 and g 2 and make their common zero set in CP k a smooth complete intersection. This perturbation of the coefficients will not change the topology of the zero locus set in RP k since before the perturbation it was a smooth manifold; the fact that the perturbation is real allows us to use Smith's theory.…”

Section: A Numerical Boundmentioning

confidence: 99%

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Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Lerario¹

2016

J. Eur. Math. Soc.

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Let X be the intersection in RP n of k quadrics, i.e. the zero locus of the homogeneous, degree two polynomials q1, . . . , q k . Let also W be the span of these polynomials in the space of all homogeneous degree two polynomials and for every r ≥ 0 let Σ (r)W equals the (spherical) intersection of W with the discriminant hypersurface in the space of quadratic polynomials; moreover for r ≥ 2 and W generic ΣWe prove that for a generic choice of q1, . . . , q k the following formula holds for the total Betti number of X:In the case we remove the nondegeneracy hypotesis the previous formula remains valid upon substitution of ΣW with a pertubation of it obtained by translating W in the direction of a small negative definite quadratic form. The previous sum (both in the generic and the general case) contains at most O(k) 1/2 summands, as the sets Σ (r) W turns out to be empty for r+1 2 ≥ k. We study the topology of symmetric determinantal varieties, like the above Σ (r) W , and bound their homological complexities, with particular interest at those defined on a sphere. Using formula (1) and the results on the complexity of determinantal varieties, we prove the sharp bound: b(X) ≤ O(n) k−1 thus improving Barvinok's style bounds (recall that the best previously known bound, due to Basu, has the shape O(n) 2k+2 ). AcknowledgementsThe author is grateful to S. Basu for many stimulating discussions and to A. A. Agrachev for his constant support.

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“…i − (ωq − ǫp) = i + (ǫp − ωq). In particular the Betti numbers of Ω j+1 and of its perturbation Ω n−j (ǫ) are the same, as proved in the following lemma from [16].…”

Section: Transversality Argumentssupporting

confidence: 52%

Section: Examplesmentioning

confidence: 98%

Section: Transversality Argumentsmentioning

confidence: 99%

Section: A Numerical Boundmentioning

confidence: 99%

See 2 more Smart Citations

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Lerario¹

2016

J. Eur. Math. Soc.

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“…Using the same technique, it is possible to prove that if X is the intersection of three real quadrics in RP n , then b(X) ≤ n(n + 1). The interested reader is referred to [16].…”

Section: B(x) ≤ 2nmentioning

confidence: 99%

Convex Pencils of Real Quadratic Forms

Lerario

2012

Discrete Comput Geom

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Abstract. We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S n ; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X) ≤ 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b k (X) ≤ 2(k + 2).

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Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications

Basu

Rizzie

2017

Discrete Comput Geom

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Abstract. We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known. Finally, we extend a result of Barone and Basu on bounding the number of connected components of real varieties defined by two polynomials of differing degrees to the sum of all Betti numbers, thus making progress on an open problem posed in that paper.

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The total Betti number of the intersection of three real quadrics

Cited by 6 publications

References 17 publications

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Convex Pencils of Real Quadratic Forms

Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications

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