Systems of quadratic inequalities

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of qua- dratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the dierential d_2

How many geodesics join two points on a contact sub-Riemannian manifold?

We investigate the structure and the topology of the set of geodesics (critical points for the energy functional) between two points on a contact Carnot group G (or, more generally, corank-one Carnot groups). Denoting by (x, z) 08 R2n 7 R exponential coordinates on G, we find constants C1, C2 > 0 and R1,R2 such that the number (Formula Presented) of geodesics joining the origin with a generic point p = (x, z) satisfies: (Formula Presented) We give conditions for p to be joined by a unique geodesic and we specialize our computations to standard Heisenberg groups, where (Formula Presented) The set of geodesics joining the origin with p 60 p0, parametrized with their initial covector, is a topological space \u393(p), that naturally splits as the disjoint union (Formula Presented) where \u3930(p) is a finite set of isolated geodesics, while \u393 1e(p) contains continuous families of non-isolated geodesics (critical manifolds for the energy). We prove an estimate similar to (1) for the \u201ctopology\u201d (i.e. the total Betti number) of \u393(p), with no restriction on p. When G is the Heisenberg group, families appear if and only if p is a vertical nonzero point and each family is generated by the action of isometries on a given geodesic. Surprisingly, in more general cases, families of non-isometrically equivalent geodesics do appear. If the Carnot group G is the nilpotent approximation of a contact sub-Riemannian manifold M at a point p0, we prove that the number \u3bd(p) of geodesics in M joining p0 with p can be estimated from below with (Formula Presented) (p). The number \u3bd(p) estimates indeed geodesics whose image is contained in a coordinate chart around p0 (we call these \u201clocal\u201d geodesics). As a corollary we prove the existence of a sequence (Formula Presented) such that: (Formula Presented)i.e. the number of \u201clocal\u201d geodesics between two points can be arbitrarily large, in sharp contrast with the Riemannian case

On the Number of Flats Tangent to Convex Hypersurfaces in Random Position

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287\u2013300, 2006; B\ufcrgisser and Lerario, in J Reine Angew Math, https://doi.org/10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617\u2013644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543\u2013571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815\u20134829, 2002; Proc Am Math Soc 133(10):2835\u20132844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by \u201cconvex hypersurfaces\u201d we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces X1,\u2026,Xdk,n 82RPn, where dk,n= (k+ 1) (n- k) , are in random position if each one of them is randomly translated by elements g1,\u2026,gdk,n sampled independently from the orthogonal group with the uniform distribution. Denoting by \u3c4k(X1,\u2026,Xdk,n) the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that \u3c4k(X1,\u2026,Xdk,n)=\u3b4k,n\ub7 0fi=1dk,n|\u3a9k(Xi)||Sch(k,n)|,where \u3b4k,n is the expected degree from [6] (the average number of k-flats incident to dk,n many random (n- k- 1) -flats), | Sch (k, n) | is the volume of the Special Schubert variety of k-flats meeting a fixed (n- k- 1) -flat (computed in [6]) and | \u3a9 k(X) | is the volume of the manifold \u3a9 k(X) 82 G(k, n) of all k-flats tangent to X. We give a formula for the evaluation of | \u3a9 k(X) | in terms of some curvature integral of the embedding X\u21aa RP n and we relate it with the classical notion of intrinsic volumes of a convex set: |\u3a9k( 02C)||Sch(k,n)|=4Vn-k-1(C),k=0,\u2026,n-1.As a consequence we prove the universal upper bound: \u3c4k(X1,\u2026,Xdk,n) 64\u3b4k,n\ub74dk,n.Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case k= 1 , n= 3 for every m> 0 we can provide examples of smooth convex hypersurfaces X1, \u2026 , X4 such that the intersection \u3a9 1(X1) 29 ef 29 \u3a9 1(X4) 82 G(1 , 3) is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions

Geodesics and horizontal-path spaces in Carnot groups

We study the topology of horizontal-paths spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the paths space). These geodesics typically appear in families (critical manifolds). Letting the energy grow, we obtain an upper bound on the number of critical manifolds with energy bounded by s: this upper bound is polynomial in s of degree l (the corank of the distribution). Despite this evidence, we show that Morse-Bott inequalities are far from sharp: the topology (i.e. the sum of the Betti numbers) of the loop space filtered by the energy grows at most as a polynomial in s of degree l-1. In the limit for s at infinity, all Betti numbers (except the zeroth) must actually vanish: the admissible-loop space is contractible. In the case the corank l=2 we compute exactly the leading coefficient of the sum of the Betti numbers of the admissible-loop space with energy less than s. This coefficient is expressed by an integral on the unit circle depending only on the coordinates of the final point and the structure constants of the Lie algebra of G

Zeroes of polynomials on definable hypersurfaces: Pathologies exist, but they are rare

Given a sequence {Zd}d?N of smooth and compact hypersurfaces in Rn-1, we prove that (up to extracting subsequences) there exists a regular definable hypersurface ? RPn such that each manifold Zd is diffeomorphic to a component of the zero set on of some polynomial of degree d. (This is in sharp contrast with the case when is semialgebraic, where for example the homological complexity of the zero set of a polynomial p on is bounded by a polynomial in deg(p).) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface ? RPn containing a subset D homeomorphic to a disk, and a family of polynomials {pm}m?N of degree deg(pm) = dm such that (D, Z(pm)nD) ~ (Rn-1, Zdm ), i.e. the zero set of pm in D is isotopic to Zdm in Rn-1. This says that, up to extracting subsequences, the intersection of with a hypersurface of degree d can be as complicated as we want. We call these 'pathological examples'. In particular, we show that for every 0 = k = n - 2 and every sequence of natural numbers a = {ad}d?N there is a regular, compact semianalytic hypersurface ? RPn, a subsequence {adm }m?N and homogeneous polynomials {pm}m?N of degree deg(pm) = dm such that bk( n Z(pm)) = adm . (0.1) (Here bk denotes the kth Betti number.) This generalizes a result of Gwozdziewicz et al. [13]. On the other hand, for a given definable we show that the Fubini-Study measure, in the Gaussian probability space of polynomials of degree d, of the set dm,a, of polynomials verifying (0.1) is positive, but there exists a constant c such tha

Quantitative Singularity Theory for Random Polynomials

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-W singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-W singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset W of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-W singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function p we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center p⁠, in which the W-singularity structure is constant

Random fields and the enumerative geometry of lines on real and complex hypersurfaces

We introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces. This can be considered as a further step in the original Shub\u2013Smale program of studying the real zeros of random polynomial systems. Our technique is general, and it also applies, for example, to the case of the enumerative geometry of flats on complete intersections. We derive a formula expressing the average number En of real lines on a random hypersurface of degree 2 n- 3 in RP n in terms of the expected modulus of the determinant of a special random matrix. In the case n= 3 we prove that the average number of real lines on a random cubic surface in RP 3 equals: E3=62-3.This technique can also be applied to express the number Cn of complex lines on a generic hypersurface of degree 2 n- 3 in CP n in terms of the expectation of the square of the modulus of the determinant of a random Hermitian matrix. As a special case, we recover the classical statement C3= 27. We determine, at the logarithmic scale, the asymptotic of the quantity En, by relating it to Cn (whose asymptotic has been recently computed in [19]). Specifically we prove that: limn\u2192 1elogEnlogCn=12.Finally we show that this approach can be used to compute the number Rn= (2 n- 3) ! ! of real lines, counted with their intrinsic signs (as defined in [28]), on a generic real hypersurface of degree 2 n- 3 in RP n

Topologies of nodal sets of random band limited functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.Comment: 62 pages. Major revision following referee repor

Random spectrahedra

Spectrahedra are affine-linear sections of the cone Pn of positive semidefinite symmetric n 7 n-matrices. We consider random spectrahedra that are obtained by intersecting Pn with the affine-linear space 1 + V , where 1 is the identity matrix and V is an `-dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of `-planes in the space of n 7 n real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for ` = 3 we relate the average number E\u3c3n of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra (n = 4) we show that E\u3c34 = 6 12 1a43 . Moreover, we prove that the average number E \u3c1n of singular points on the real variety of singular matrices in 1 + V is n(n 12 1). This quantity is related to the volume of the variety of real symmetric matrices with repeated eigenvalues. Furthermore, we compute the asymptotics of the volume and the volume of the boundary of a random spectrahedron