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Antonio Lerario

Publications and source records attributed to Antonio Lerario.

At least 55 records · Page 3Linked to original sources
arXiv2018

Low degree approximation of random polynomials

We prove that with "high probability" a random Kostlan polynomial in $n+1$ many variables and of degree $d$ can be approximated by a polynomial of "low degree" without changing the topology of its zero set on the sphere $S^n$. The dependence between the "low degree" of the approximation and the "high probability" is quantitative: for example, with overwhelming probability the zero set of a Kostlan polynomial of degree $d$ is isotopic to the zero set of a polynomial of degree $O(\sqrt{d \log d})$. The proof is based on a probabilistic study of the size of $C^1$-stable neighborhoods of Kostlan polynomials. As a corollary we prove that certain topological types (e.g. curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.

math.AG↗
arXiv2018

Homotopy properties of horizontal loop spaces and applications to closed sub-riemannian geodesics

Given a manifold $M$ and a proper sub-bundle $Δ\subset TM$, we study homotopy properties of the horizontal base-point free loop space $Λ$, i.e. the space of absolutely continuous maps $γ:S^1\to M$ whose velocities are constrained to $Δ$ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map $F:Λ\to M$ (the map associating to every loop its base-point) is a Hurewicz fibration for the $W^{1,2}$ topology on $Λ$. Using this result we show that, even if the space $Λ$ might have deep singularities (for example: constant loops form a singular manifold homeomorphic to $M$), its homotopy can be controlled nicely. In particular we prove that $Λ$ (with the $W^{1,2}$ topology) has the homotopy type of a CW-complex, that its inclusion in the standard base-point free loop space (i.e. the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently its homotopy groups can be computed as $π_k(Λ)\simeq π_k(M) \ltimes π_{k+1}(M)$ for all $k\geq 0.$ These topological results are applied, in the second part of the paper, to the problem of the existence of closed sub-riemannian geodesics. In the general case we prove that if $(M, Δ)$ is a compact sub-riemannian manifold, each non trivial homotopy class in $π_1(M)$ can be represented by a closed sub-riemannian geodesic. In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if $(M, Δ)$ is a compact, contact manifold, then every sub-riemannian metric on $Δ$ carries at least one closed sub-riemannian geodesic. This result is based on a combination of the above topological results with a delicate study of the Palais-Smale condition in the vicinity of abnormal loops (singular points of $Λ$).

math.DG↗
arXiv2018

On the geometry of the set of symmetric matrices with repeated eigenvalues

We investigate some geometric properties of the real algebraic variety $Δ$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $Δ$. We exhibit connections of our study to Real Algebraic Geometry (computing the Euclidean Distance Degree of $Δ$) and Random Matrix Theory.

math.AG↗
arXiv2018

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

Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in $\mathbb{R}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $Γ\subset \mathbb{R}\mathrm{P}^n$ such that each manifold $Z_d$ appears as a component of the zero set on $Γ$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $Γ$ is algebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $Γ$ is bounded by a polynomial in $\mathrm{deg}(p)$.) We call these "pathological examples". In particular, we show that for every $0 \leq k \leq n-2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact and definable hypersurface $Γ\subset \mathbb{R}\mathrm{P}^n$, a subsequence $\{a_{d_m}\}_{m\in \mathbb{N}}$ and homogeneous polynomials $\{p_{m}\}_{m\in \mathbb{N}}$ of degree $\mathrm{deg}(p_m)=d_m$ such that: \begin{equation} \label{eq:pathintro} b_k(Γ\cap Z(p_m))\geq a_{d_m}.\end{equation} (Here $b_k$ denotes the $k$-th Betti number.) This generalizes a result of Gwoździewicz, Kurdyka and Parusiński. On the other hand, for a given definable $Γ$ we show that the Fubini-Study measure, in the gaussian space of polynomials of degree $d$, of the set $Σ_{d_m,a, Γ}$ of polynomials verifying $b_k(Γ\cap Z(p_m))\geq a_{d_m}$ is positive, but there exists a contant $c_Γ$ such that this measure can be bounded by: \begin{equation} 0<\mathbb{P}(Σ_{d_m, a, Γ})\leq \frac{c_Γ d_m^{\frac{n-1}{2}}}{a_{d_m}}. \end{equation} This shows that the set of "pathological examples" has "small" measure.

math.AG↗
arXiv2018

Probabilistic Schubert Calculus

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\textrm{edeg} G(k,n)$ of the real Grassmannian $G(k,n)$ as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random subspaces of $\mathbb{R}^n$, all of dimension $n-k$, where these subspaces are sampled uniformly and independently from $G(n-k,n)$. We express $\textrm{edeg} G(k,n)$ in terms of the volume of an invariant convex body in the tangent space to the Grassmanian, and prove that for fixed $k\ge 2$ and $n\to\infty$, $$ \textrm{edeg} G(k,n) = \textrm{deg} G_\mathbb{C}(k,n)^{\frac{1}{2} ε_k + o(1)}, $$ where $\textrm{deg} G_\mathbb{C}(k,n)$ denotes the degree of the corresponding complex Grassmannian and $ε_k$ is monotonically decreasing with $\lim_{k\to\infty} ε_k = 1$. In the case of the Grassmannian of lines, we prove the finer asymptotic \begin{equation*} \textrm{edeg} G(2,n+1) = \frac{8}{3π^{5/2}\sqrt{n}}\, \left(\frac{π^2}{4} \right)^n \left(1+\mathcal{O}(n^{-1})\right). \end{equation*} The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set $X\subseteq\mathbb{R}\textrm{P}^{n-1}$ of dimension $n-k-1$ its Chow hypersurface $Z(X)\subseteq G(k,n)$, consisting of the $k$-planes $A$ in $\mathbb{R}^n$ whose projectivization intersects $X$. Denoting $N:=k(n-k)$, we show that $$ \mathbb{E}\#\left(g_1Z(X_1)\cap\cdots\cap g_N Z(X_N)\right) = \textrm{edeg} G(k,n) \cdot \prod_{i=1}^{N} \frac{|X_i|}{|\mathbb{R}\textrm{P}^{m}|}, $$ where each $X_i$ is of dimension $m=n-k-1$, the expectation is taken with respect to independent uniformly distributed $g_1,\ldots,g_m\in O(n)$ and $|X_i|$ denotes the $m$-dimensional volume of $X_i$.

math.AG↗
arXiv2018

On the number of flats tangent to convex hypersurfaces in random position

We investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a random point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, where $d_{k,n}=(k+1)(n-k)$, are in random position if each one of them is randomly translated by elements $g_1, \ldots, g_{d_{k,n}}$ sampled independently and uniformly from the Orthogonal group; we denote by $τ_k(X_1, \ldots, X_{d_{k,n}})$ the average number of $k$-dimensional projective subspaces (flats) which are simultaneously tangent to all the hypersurfaces. We prove that $$τ_k(X_1, \ldots, X_{d_{k,n}})=δ_{k,n}\cdot\prod_{i=1}^{d_{k,n}}\frac{|Ω_k(X_i)|}{|\textrm{Sch}(k,n)|},$$ where $δ_{k,n}$ is the expected degree (the average number of $k$-flats incident to $d_{k,n}$ many random $(n-k-1)$-flats), $|\textrm{Sch}(k,n)|$ is the volume of the Special Schubert variety of $k$-flats meeting a $(n-k-1)$-flat and $|Ω_k(X)|$ is the volume of the manifold of all $k$-flats tangent to $X$. We give a formula for the evaluation of $|Ω_k(X)|$ in term of some curvature integral of the embedding $X\hookrightarrow \mathbb{R}\textrm{P}^n$ and we relate it with the notion of intrinsic volumes of a convex set: $$\frac{|Ω_{k}(\partial C)|}{|\textrm{Sch}(k, n)|}=4|V_{n-k-1}(C)|,\quad k=0,\ldots,n-1.$$ We prove the upper bound: $$τ_k(X_1,\ldots,X_{d_{k,n}})\leqδ_{k, n}\cdot 4^{d_{k,n}}.$$ In the case $k=1,n=3$ for every $m>0$ we provide examples of smooth convex hypersurfaces $X_1,\ldots,X_4$ such that the intersection $Ω_1(X_1)\cap\cdots\capΩ_1(X_4)\subset\mathbb{G}(1,3)$ is transverse and consists of at least $m$ lines. We also present analogous results for semialgebraic hypersurfaces satisfying some nondegeneracy assumption.

math.AG↗
arXiv2017

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

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z) \in \mathbb{R}^{2n} \times \mathbb{R}$ centered at $p$. In this case we prove that for the generic $q$ the number of geodesics $ν(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{|z|}{\|x\|^2} + R_1 \leq ν(q) \leq C_2\frac{|z|}{\|x\|^2} + R_2\] for some constants $C_1,C_2$ and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2 = \frac{8}π$. Removing the genericity condition for $q$, geodesics might appear in families and we prove a similar statement for their topology. We study these families, and in particular we focus on the unexpected appearance of isometrically non-equivalent geodesics: families on which the action of isometries is not transitive. We apply the previous study to contact sub-Riemannian manifolds: we prove that for any given point $p \in M$ there is a sequence of points $p_n$ such that $p_n \to p$ and that the number of geodesics between $p$ and $p_n$ grows unbounded (moreover these geodesics have the property of being contained in a small neighborhood of $p$).

math.DG↗
arXiv2016

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

We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$ 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 $\mathbb{R}\textrm{P}^3$ equals: $$E_3=6\sqrt{2}-3.$$ Our technique can also be used to express the number $C_n$ of complex lines on a generic hypersurface of degree $2n-3$ in $\mathbb{C}\textrm{P}^n$ in terms of the determinant of a random Hermitian matrix. As a special case we obtain a new proof of the classical statement $C_3=27.$ We determine, at the logarithmic scale, the asymptotic of the quantity $E_n$, by relating it to $C_n$ (whose asymptotic has been recently computed D. Zagier). Specifically we prove that: $$\lim_{n\to \infty}\frac{\log E_n}{\log C_n}=\frac{1}{2}.$$ Finally we show that this approach can be used to compute the number $R_n=(2n-3)!!$ of real lines, counted with their intrinsic signs, on a generic real hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$.

math.AG↗
arXiv2016

On the zeros of random harmonic polynomials: the truncated model

A probabilistic approach to the study of the number of zeros of complex harmonic polynomials was initiated by W. Li and A. Wei (2009), who derived a Kac-Rice type formula for the expected number of zeros of random harmonic polynomials with independent Gaussian coefficients. They also provided asymptotics for a complex version of the Kostlan ensemble. Here we determine asymptotics for the alternative truncated model that was recently proposed by J. Hauenstein, D. Mehta, and the authors. Our results confirm (and sharpen) their (3/2)-powerlaw conjecture that had been formulated on the basis of computer experiments.

math.CV↗
arXiv2016

Homotopically Invisible Singular Curves

Given a smooth manifold $M$ and a totally nonholonomic distribution $Δ\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are critical points of the endpoint map $F:γ\mapstoγ(1)$ defined on the space $Ω$ of horizontal paths starting at a fixed point $x$. We consider a subriemannian energy $J:Ω(y)\to\mathbb R$, where $Ω(y)=F^{-1}(y)$ is the space of horizontal paths connecting $x$ with $y$, and study those singular paths that do not influence the homotopy type of the Lebesgue sets $\{γ\inΩ(y)\,|\,J(γ)\le E\}$. We call them homotopically invisible. It turns out that for $d\geq 3$ generic subriemannian structures have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a subriemannian Minimax principle and discuss some applications).

math.DG↗
arXiv2016

On the geometry of random lemniscates

We investigate the geometry of a random rational lemniscate $Γ$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational functions $r=p/q$ of degree $n$ by sampling $p$ and $q$ independently from the complex Kostlan ensemble of random polynomials of degree $n$. We prove that the average \emph{spherical length} of $Γ$ is $\frac{π^2}{2} \sqrt{n},$ which is proportional to the square root of the maximal spherical length. We also provide an asymptotic for the average number of points on the curve that are tangent to one of the meridians on the Riemann sphere (i.e. tangent to one of the radial directions in the plane). Concerning the topology of $Γ$, on a local scale, we prove that for every disk $D$ of radius $O(n^{-1/2})$ in the Riemann sphere and any \emph{arrangement} (i.e. embedding) of finitely many circles $A\subset D$ there is a positive probability (independent of $n$) that $(D,Γ\cap D)$ is isotopic to $( D,A)$. (A local random version of Hilbert's Sixteenth Problem restricted to lemniscates.) Corollary: the average number of connected components of $Γ$ increases linearly (the maximum rate possible according to a deterministic upper bound).

math.CV↗
arXiv2015

On the number of connected components of random algebraic hypersurfaces

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d \rightarrow \infty$), we prove that the order of growth of $\mathbb{E} b_0(X)$ satisfies: $$\mathbb{E} b_{0}(X)=Θ\left(\left[ \mathbb{E} b_0(X\cap \mathbb{R}P^1) \right]^{n} \right). $$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f|_{\mathbb{R}P^1}.$ The proof requires an upper bound for $\mathbb{E} b_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E} b_0(X)$ (as $n \rightarrow \infty$).

math.AG↗
arXiv2015

Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the number of their critical points is infinite (we use Lusternik-Schnirelmann category combined with the Hurewicz property). In the special case where the control system is subriemannian this result can be read as the corresponding version of Serre's theorem, on the existence of infinitely many geodesics between two points on a compact riemannian manifold. In the subriemannian case we show that the Hurewicz property holds for all $p\geq1$ and the horizontal-loop space with the $W^{1,2}$ topology has the homotopy type of a CW-complex (as long as the endpoint map has at least one regular value); in particular the inclusion of the horizontal-loop space in the ordinary one is a homotopy equivalence.

math.DG↗
arXiv2014

Experiments on the zeros of harmonic polynomials using certified counting

Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale's alpha-theory to certify the results. Using this approach, we provide new examples of harmonic polynomials having the most extreme number of zeros known so far; we also study the mean and variance of the number of zeros of random harmonic polynomials.

math.CV↗
arXiv2014

Intrinsic volumes of set of singular matrices

We explicitly compute the intrinsic volume of the set of real (and real symmetric) matrices of Frobenius norm one and given corank (the case of matrices with zero determinant as a special case). We give asymptotic formulas for our computations and we discuss several examples and applications.

math.DG↗
arXiv2013

Gap probabilities and applications to geometry and random topology

We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a first application, we consider the set of n x n (Real, Complex or Quaternionic) Hermitian matrices with Frobenius norm one and determinant zero. We give an exact formula for the intrinsic volume of this set and as n goes to infinity its asymptotic (with an explicit constant) is of the order n^(1/2). As a second application we consider the problem of computing Betti numbers of an intersection of k random Kostlan quadrics in RP^n. We show that the i-th Betti number is asymptotically expected to be one (for i sufficiently away from n/2). In the case k=2 the the sum of all Betti numbers was recently shown by the first author to equal n+o(n). Here we sharpen this asymptotic proving an asymptotic with two orders of precision and explicit constants.

math.PR↗
arXiv2013

Remarks on Wilmshurst's theorem

We demonstrate counterexamples to Wilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded then it must have codimension at least two. Examples are provided to show that this conclusion cannot be improved.

math.CV↗