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

Abstract

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.

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

“…For projective varieties in P k R defined by a fixed number of homogeneous quadratic polynomials we have the following bound that is asymptotically a slight improvement over the tightest bound known previously [36,Theorem 15] (namely, the bound (O(k)) −1 ). Theorem 17.…”

“…This bound was later sharpened in [14] and further sharpened in the case of algebraic sets by Lerario in [36,Theorem 15], where the following nearly optimal result was proved.…”

“…Remark 5. The main tool used in the proof of Theorem 8 was a technique introduced by Agrachev in [5,2,3] and later exploited by several authors [15,4,35,36] for bounding the Betti numbers of semi-algebraic sets defined by quadratic polynomials.…”

“…a constant number of) quadratic equalities and inequalities. Barvinok [10] proved a polynomial bound on the Betti numbers of semi-algebraic sets (see Theorem 4 for a precise statement), which were sharpened in [14,36], and also extended to a more general setting in [15] (see Theorem 8 for a precise statement). Some of these results were proved using different techniques than a simple counting of critical points -for example, a spectral sequence argument first proposed by Agrachev [3,2,5] plays an important role in the results proved in [15,36].…”

Geodesics and horizontal-path spaces in Carnot groups

“…Associating to each v ∈ E(ω) the control e tωA v defines a linear injection of E(ω) into H = ⊕ k≥1 T k ; in particular the previous curve admits the Fourier series decomposition (14) e…”