arXiv subjects

Jiří Lebl

Publications and source records attributed to Jiří Lebl.

8 recordsLinked to original sources

arXiv2026

Hermitian rank in ideal powers

We prove that the (hermitian) rank of $QP^d$ is bounded from below by the rank of $P^d$ whenever $Q$ is not identically zero and real-analytic in a neighborhood of some point on the zero set of $P$ in $\mathbb{C}^n$ and $P$ is a polynomial of bidegree at most $(1,1)$. This result generalizes the theorem of D'Angelo and the second author for bihomogeneous $P$ and Huang's Lemma. The inequality demonstrates that real-analytic multipliers cannot reduce the algebraic complexity of ideal powers. Examples show that no hypothesis can be dropped. As an application, we give a lower bound on the target dimension for a CR map of hyperquadrics depending on the order of contact.

math.CV↗

arXiv2024

Records from the S-Matrix Marathon: Tasty Bits of Several Complex Variables

We will cover the basics of several complex variables in 4 lectures: Basic properties of holomorphic functions in several variables, the notion of pseudoconvexity, CR functions and CR geometry, and the $\bar\partial$-problem. The main underlying idea is to connect various characterizations of domains of holomorphy, that is, the natural domains of definition for holomorphic functions. In the process we will connect the function theory on the domain to geometric properties of the boundary, and discuss the relationship between the boundary values and the functions themselves and extension of holomorphic functions from subspaces. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11-22 March 2024.

hep-th↗

arXiv2024

Proper maps of ball complements & differences and rational sphere maps

We consider proper holomorphic maps of ball complements and differences in complex euclidean spaces of dimension at least two. Such maps are always rational, which naturally leads to a related problem of classifying rational maps taking concentric spheres to concentric spheres, what we call $m$-fold sphere maps; a proper map of the difference of concentric balls is a $2$-fold sphere map. We prove that proper maps of ball complements are in one to one correspondence with polynomial proper maps of balls taking infinity to infinity. We show that rational $m$-fold sphere maps of degree less than $m$ (or polynomial maps of degree $m$ or less) must take all concentric spheres to concentric spheres and we provide a complete classification of them. We prove that these degree bounds are sharp.

math.CV↗

arXiv2024

On Segre-degenerate Levi-flat hypervarieties

We prove that a singular real-analytic Levi-flat hypersurface $H$ in $\mathbb C^n$ being Segre-degenerate at a point $p$ is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects $H$ at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of $H$ at $p$. The existence of such two-sided support provides families of analytic discs attached to $H$ that covers a neighborhood of $p$. The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to a fixed neighborhood of $p$. Second, the rational hull of $H$ is a neighborhood of $p$, and thus no Levi-flat Segre-degenerate hypersurface in $\mathbb C^n$ can be rationally convex.

math.CV↗

arXiv2024

Rational Maps of Balls and their Associated Groups

Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we also provide several new groups associated to the map.

math.CV↗

arXiv2021

On CR singular CR images

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case removability is equivalent to $M$ being the image of a generic real-analytic submanifold $N$ under a holomorphic map that is a diffeomorphism of $N$ onto $M$, what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

math.CV↗

arXiv2021

Variational Quantum Algorithms for Euclidean Discrepancy and Covariate-Balancing

Algorithmic discrepancy theory seeks efficient algorithms to find those two-colorings of a set that minimize a given measure of coloring imbalance in the set, its {\it discrepancy}. The {\it Euclidean discrepancy} problem and the problem of balancing covariates in randomized trials have efficient randomized algorithms based on the Gram-Schmidt walk (GSW). We frame these problems as quantum Ising models, for which variational quantum algorithms (VQA) are particularly useful. Simulating an example of covariate-balancing on an IBM quantum simulator, we find that the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) yield results comparable to the GSW algorithm.

quant-ph↗

arXiv2018

Removing Isolated Zeroes by Homotopy

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion of index for vector fields, the $q=n$ case. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call "locally inessential," and for dimensions $n$, $q$ where $π_{n-1}(S^{q-1})$ is trivial, every isolated zero is locally inessential. We consider the problem of constructing such an approximation $g$, and show that there exists a continuous homotopy from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists such a homotopy which is also semialgebraic. For $q=2$ and $f$ real analytic with a locally inessential isolated zero, there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

math.GT↗