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Jiri Lebl

Publications and source records attributed to Jiri Lebl.

At least 19 recordsLinked to original sources
arXiv2026

Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms

We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \[ g_\sigma(z)=1+\sigma_1 z_1^2+\sigma_2 z_2^2, \qquad \sigma_1,\sigma_2\geq 0. \] A basic question is: which pairs $(\sigma_1,\sigma_2)$ can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if \[ 0\leq \sigma_1,\sigma_2<1, \qquad \sqrt{1-\sigma_1^2}+\sqrt{1-\sigma_2^2}>1. \] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ \sigma=(\sigma_1,\sigma_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_\sigma$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $\sigma$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_\sigma$. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.

math.CV↗
arXiv2025

Proper maps of annuli

We study proper holomorphic maps of annuli in complex Euclidean spaces, that is, domains with $U(n)$ as the automorphism group. By the Hartogs phenomenon and a result of Forstneri\v{c}, such maps are always rational and extend to proper maps of balls. We first prove that a proper map of annuli from $n$ dimensions to $N$ dimensions where $N < \binom{n+1}{2}$ is always an affine embedding. This inequality is sharp as the homogeneous map of degree 2 satisfies $N=\binom{n+1}{2}$. Next we find a necessary and sufficient condition for a map to be homogeneous: A proper map of annuli is homogeneous if and only if its general hyperplane rank, the affine dimension of the image of a general hyperplane, is exactly $N-1$. As a corollary, we obtain a classification of homogeneous proper maps of balls. A homogeneous proper ball map takes all spheres centered at the origin to spheres centered at the origin. We show that if a proper ball map has general hyperplane rank $N-1$ and takes one sphere centered at the origin to a sphere centered at the origin, then it is homogeneous. Another corollary of this result is a complete classification of proper maps of annuli from dimension 2 to dimension 3. Finally, we give a complete normal form of rational proper maps of annuli of degree 2.

math.CV↗
arXiv2024

Intrinsic complexification of real-analytic varieties

We introduce a framework for Segre varieties for singular real-analytic subvarieties of a complex space and utilize it to study the intrinsic complexifications of these subvarieties. Many examples illustrate the subtle issues arising in the singular setting.

math.CV↗
arXiv2024

CR functions at CR singularities: approximation, extension, and hulls

We study three possible definitions of the notion of CR functions at CR singular points, their extension to a fixed-neighborhood of the singular point, and analogues of the Baouendi--Tr\`eves approximation in a fixed neighborhood. In particular, we give a construction of certain disc hulls, which, if large enough, give the fixed-neighborhood extension and approximation properties. We provide many examples showing the distinctions between the classes and the various properties studied.

math.CV↗
arXiv2022

Exhaustion functions and normal forms for proper maps of balls

We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials.

math.CV↗
arXiv2022

Integer Sequences and Output Arrays

The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related Catalan numbers, sequence A000108 in OEIS. We therefore introduce a general construction that places these sequences on the same footing and suggests many new integer sequences. We provide code for performing this construction and a lengthy list of examples. This construction determines a function from input sequences to output sequences. Some of the resulting output sequences are in OEIS and others are not.

math.NT↗
arXiv2021

Cartan uniqueness theorem on nonopen sets

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties satisfying the condition, the only automorphisms, CR or infinitesimal, are linear.

math.CV↗
arXiv2021

Segre-Degenerate Points Form a Semianalytic Set

We prove that the set of Segre-degenerate points of a real-analytic subvariety $X$ in ${\mathbb{C}}^n$ is a closed semianalytic set. It is a subvariety if $X$ is coherent. More precisely, the set of points where the germ of the Segre variety is of dimension $k$ or greater is a closed semianalytic set in general, and for a coherent $X$, it is a real-analytic subvariety of $X$. For a hypersurface $X$ in ${\mathbb{C}}^n$, the set of Segre-degenerate points, $X_{[n]}$, is a semianalytic set of dimension at most $2n-4$. If $X$ is coherent, then $X_{[n]}$ is a complex subvariety of (complex) dimension $n-2$. Example hypersurfaces are given showing that $X_{[n]}$ need not be a subvariety and that it also needs not be complex; $X_{[n]}$ can, for instance, be a real line.

math.CV↗
arXiv2021

An example of a compact non-C-analytic real subvariety of ${\mathbb R}^3$

The purpose of this short expository note is to provide an example exhibiting some of the pathological properties of real-analytic subvarieties, where the pathology can be visualized, and the proofs use only elementary properties of analytic functions. We construct a compact irreducible real-analytic subvariety $S$ of ${\mathbb R}^3$ of pure dimension two such that 1) the only a real-analytic function is defined in a neighbourhood of $S$ and vanishing on $S$ is the zero function, 2) the singular set of $S$ is not a subvariety of $S$, nor is it contained in any one-dimensional subvariety of $S$, 3) the variety $S$ contains a proper subvariety of dimension two. The example shows how a badly behaved part of a subvariety can be hidden via a second well-behaved component to create a subvariety of a larger set. The pathology is visualized using several figures. Examples of these phenomena are known since the time of Cartan, but hard to find in the English language literature.

math.AG↗
arXiv2021

A CR singular analogue of Severi's theorem

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order.

math.CV↗
arXiv2020

Segre nondegenerate totally real subvarieties

We study an irreducible real-analytic germ of an $n$-dimensional variety in $n$ dimensional complex space. Assuming that the variety is Segre nondegenerate we define an averaging operator that generalizes the Moser--Webster involution. This operator can be thought of as being the CR structure of the singularity, and using this operator we study the set of functions that are restrictions of holomorphic functions. We give a condition on the flattening of the singularity, that is realizing the singularity as a codimention one subvariety of a nonsingular Levi-flat hypersurface.

math.CV↗
arXiv2018

On the Levi-flat Plateau problem

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.

math.CV↗
arXiv2018

Unentangled Measurements and Frame Functions

Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author's Unentangled Gleason's Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems, and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.

quant-ph↗
arXiv2017

On Lewy extension for smooth hypersurfaces in ${\mathbb C}^n \times {\mathbb R}$

We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset {\mathbb C}^{n}\times {\mathbb R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of ${\mathbb C}^n \times {\mathbb R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.

math.CV↗
arXiv2017

Extension of CR functions from boundaries in ${\mathbb C}^n \times {\mathbb R}$

Let $Ω\subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial Ω$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial Ω\to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial Ω$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\barΩ$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial Ω$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\barΩ$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial Ω$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

math.CV↗
arXiv2017

Codimension two CR singular submanifolds and extensions of CR functions

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $Ω\subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial Ω$ has only nondegenerate CR singularities. We prove that if $f \colon \partial Ω\to {\mathbb{C}}$ is a real-analytic function that is CR at CR points of $\partial Ω$, then $f$ extends to a holomorphic function on a neighborhood of $\overlineΩ$ in ${\mathbb{C}}^n \times {\mathbb{C}}$.

math.CV↗
arXiv2016

Local Distinguishability of Generic Unentangled Orthonormal Bases

An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.

quant-ph↗