arXiv subjects

Erik Lundberg

Publications and source records attributed to Erik Lundberg.

At least 19 recordsLinked to original sources

arXiv2026

Counterexamples to Magnanini's conjecture concerning critical points of the torsion function

We construct planar simply-connected domains whose torsion function has an arbitrary prescribed number of local maxima while the distance to the boundary has only one local maximum inside the domain. This disproves a conjecture of Magnanini asserting that the number of local maxima of the torsion function is bounded above by the number of local maxima of the distance function to the boundary. We also answer a question of Steinerberger concerning the eccentricity of level sets near the point where the torsion function attains its maximum in simply-connected domains.

math.AP↗

arXiv2025

Sharp bounds for the valence of certain logharmonic polynomials

Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner.

math.CV↗

arXiv2025

On the valence of logharmonic polynomials

Investigating a problem posed by W. Hengartner (2000), we study the maximal valence (number of preimages of a prescribed point in the complex plane) of logharmonic polynomials, i.e., complex functions that take the form $f(z) = p(z) \overline{q(z)}$ of a product of an analytic polynomial $p(z)$ of degree $n$ and the complex conjugate of another analytic polynomial $q(z)$ of degree $m$. In the case $m=1$, we adapt an indirect technique utilizing anti-holomorphic dynamics to show that the valence is at most $3n-1$. This confirms a conjecture of Bshouty and Hengartner (2000). Using a purely algebraic method based on Sylvester resultants, we also prove a general upper bound for the valence showing that for each $n,m \geq 1$ the valence is at most $n^2+m^2$. This improves, for every choice of $n,m \geq 1$, the previously established upper bound $(n+m)^2$ based on Bezout's theorem. We also consider the more general setting of polyanalytic polynomials where we show that this latter result can be extended under a nondegeneracy assumption.

math.CV↗

arXiv2025

On the area of polynomial lemniscates

Erdös posed in 1940 the extremal problem of studying the minimal area of the lemniscate $\{|p(z)|<1\}$ of a monic polynomial $p$ of degree $n$ all of whose zeros are in the closed unit disc. In this article, we prove that there exist positive constants $c,C$ independent of the degree $n$ such that \[ \dfrac{c}{\log n} \leq \min \text{Area}( \{ |p(z)|<1 \} ) \leq \frac{C}{\log \log n},\] improving substantially the previously best known lower bound (due to Pommerenke in 1961) as well as improving the best known upper bound (due to Wagner in 1988). We also study the inradius (radius of the largest inscribed disc); we provide an estimate for the inradius in terms of the area that confirms a 2009 conjecture of Solynin and Williams, and we use this to give a lower bound of order $(n \sqrt{\log n})^{-1}$ on the inradius, addressing a 1958 problem posed by Erdös, Herzog, and Piranian (confirming their conjecture up to the logarithmic factor). In addition to studying the area of $\{|p(z)|<1\}$, we consider other sublevel sets $\{|p(z)| 1$ and proving power law upper and lower bounds when $0<t<1$. We also consider the minimal area problem under a more general constraint, namely, replacing the unit disc with a compact set $K$ of unit capacity, where we show that the minimal area converges to zero as $n \rightarrow \infty$ (giving an affirmative answer to another question of Erdös, Herzog, Piranian); we also investigate the structure of the area minimizing polynomials, showing that the normalized zero-counting measure converges to the equilibrium measure of $K$ as the degree $n \rightarrow \infty$.

math.CV↗

arXiv2024

The distribution of the length of the longest path in random acyclic orientations of a complete bipartite graph

Randomly sampling an acyclic orientation on the complete bipartite graph $K_{n,k}$ with parts of size $n$ and $k$, we investigate the length of the longest path. We provide a probability generating function for the distribution of the longest path length, and we use Analytic Combinatorics to perform asymptotic analysis of the probability distribution in the case of equal part sizes $n = k$ tending toward infinity. We show that the distribution is asymptotically Gaussian, and we obtain precise asymptotics for the mean and variance. These results address a question asked by Peter J. Cameron. Keywords: bipartite graph, directed graph, random graph, acyclic orientation, poly-Bernoulli numbers, lonesum matrices, generating function, analytic combinatorics, asymptotics.

math.CO↗

arXiv2023

The valence of harmonic polynomials viewed through the probabilistic lens

We prove the existence of complex polynomials $p(z)$ of degree $n$ and $q(z)$ of degree $m<n$ such that the harmonic polynomial $ p(z) + \overline{q(z)}$ has at least $\lceil n \sqrt{m} \rceil$ many zeros. This provides an array of new counterexamples to Wilmshurst's conjecture that the maximum valence of harmonic polynomials $p(z)+\overline{q(z)}$ taken over polynomials $p$ of degree $n$ and $q$ of degree $m$ is $m(m-1)+3n-2$. More broadly, these examples show that there does not exist a linear (in $n$) bound on the valence with a uniform (in $m$) growth rate. The proof of this result uses a probabilistic technique based on estimating the average number of zeros of a certain family of random harmonic polynomials.

math.CV↗

arXiv2023

On the average number of zeros of random harmonic polynomials with i.i.d. coefficients: precise asymptotics

Addressing a problem posed by W. Li and A. Wei (2009), we investigate the average number of (complex) zeros of a random harmonic polynomial $p(z) + \overline{q(z)}$ sampled from the Kac ensemble, i.e., where the coefficients are independent identically distributed centered complex Gaussian random variables. We establish a precise asymptotic, showing that when $\text{deg} p = \text{deg} q = n$ tends to infinity the average number of zeros is asymptotic to $\frac{1}{2} n \log n$. We further consider the average number of zeros restricted to various regions in the complex plane leading to interesting comparisons with the classically studied case of analytic Kac polynomials. We also consider deterministic extremal problems for harmonic polynomials with coefficient constraints; using an indirect probabilistic method we show the existence of harmonic polynomials with unimodular coefficients having at least $\frac{2}π n \log n + O(n)$ zeros. We conclude with a list of open problems.

math.CV↗

arXiv2023

Limit cycle enumeration in random vector fields

We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. Investigating a problem introduced by Brudnyi [Annals of Mathematics (2001)] we also consider a special local setting of counting limit cycles near a randomly perturbed center focus, and when the perturbation has i.i.d. coefficients, we prove a limit law showing that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a logarithmically-correlated random univariate power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global average count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.

math.DS↗

arXiv2023

Multiplane gravitational lenses with an abundance of images

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of $K$-plane point-mass gravitational lens ensembles that produce $\prod_{i=1}^K (5g_i-5)$ images of a single background source, where $g_i$ is the number of point masses in the $i^\text{th}$ plane. This gives asymptotically (for large $g_i$ with $K$ fixed) $5^K$ times the minimal number of lensed images. Our construction uses Rhie's single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from differential topology, we then show that the preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including the positions of lensed images as well as the structure of the critical curves and caustics. We observe an interesting ``caustic of multiplicity'' phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case.

math-ph↗

arXiv2023

Inradius of random lemniscates

A classically studied geometric property associated to a complex polynomial $p$ is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate $Λ:= \{z \in \mathbb{C}:|p(z)| < 1\}$. In this paper, we study the lemniscate inradius when the defining polynomial $p$ is random, namely, with the zeros of $p$ sampled independently from a compactly supported probability measure $μ$. If the negative set of the logarithmic potential $U_μ$ generated by $μ$ is non-empty, then the inradius is bounded from below by a positive constant with overwhelming probability. Moreover, the inradius has a determinstic limit if the negative set of $U_μ$ additionally contains the support of $μ$. On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then the inradius converges in distribution to a random variable taking values in $(0,1/2)$. We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we show is close to the unit disk with overwhelming probability.

math.PR↗

arXiv2022

The number of limit cycles bifurcating from a randomly perturbed center

We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with independent coefficients. This problem reduces, by way of classical perturbation theory of the Poincaré first return map, to a problem on the real zeros of a random \emph{univariate} polynomial $\displaystyle f_n(x) = \sum_{m=0}^n c_m ξ_m x^m$ with independent coefficients $ξ_m$ having mean zero, variance 1 and $c_m \sim m^{-1/2}$. This polynomial belongs to the class of {\it generalized Kac polynomials} at the critical regime. We provide asymptotics for the average number of real zeros and answer the question on bifurcating limit cycles. Additionally, we provide the correct order of the mean number of real roots in the subcritical regime.

math.PR↗

arXiv2021

Homotopy Types of Random Cubical Complexes

We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability $p$, and our results show a qualitative change in the homotopy measure as $p$ crosses the percolation threshold $p=p_c$. Specializing to the case of $d=2$ dimensions, we also present empirical results that raise further questions on the $p$-dependence of the limiting homotopy measure.

math.PR↗

arXiv2021

A note on arclength null quadrature domains

We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.

math.CV↗

arXiv2021

On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force

We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted $n$-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by $n+1$, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in $n$. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as ``ring configurations'', consisting of $n-1$ equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as $5n-5$ equilibria. We verify analytically that the ring configuration has at least $5n-5$ equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing.

math-ph↗

arXiv2021

A note on the critical points of the localization landscape

Let $Ω\subset\mathbb{C}$ be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function $v=v_Ω$ that solves the elliptic problem $Δv = -2$ in $Ω,$ with boundary values $v=0$ on $\partialΩ.$ This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of $v$ when $Ω$ belongs to a special class of domains in the plane, namely, domains for which the boundary $\partialΩ$ is contained in $\{z:|z|^2 = f(z) + \overline{f(z)}\},$ where $f'(z)$ is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when $Ω$ is a quadrature domain, and conclude the note by stating some open problems and conjectures.

math.CV↗

arXiv2020

Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit

We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called "thermodynamic" regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean geometric complexes of the same dimension as the manifold.

math.PR↗

arXiv2020

Asymptotic enumeration of lonesum matrices

We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and we strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.

math.CO↗

arXiv2018

EGBTER: Capturing degree distribution, clustering coefficients, and community structure in a single random graph model

Random graph models are important constructs for data analytic applications as well as pure mathematical developments, as they provide capabilities for network synthesis and principled analysis. Several models have been developed with the aim of faithfully preserving important graph metrics and substructures. With the goal of capturing degree distribution, clustering coefficient, and communities in a single random graph model, we propose a new model to address shortcomings in a progression of network modeling capabilities. The Block Two-Level Erd{ő}s-R{é}nyi (BTER) model of Seshadhri et al., designed to allow prescription of expected degree and clustering coefficient distributions, neglects community modeling, while the Generalized BTER (GBTER) model of Bridges et al., designed to add community modeling capabilities to BTER, struggles to faithfully represent all three characteristics simultaneously. In this work, we fit BTER and two GBTER configurations to several real-world networks and compare the results with that of our new model, the Extended GBTER (EGBTER) model. Our results support that EBGTER adds a community-modeling flexibility to BTER, while retaining a satisfactory level of accuracy in terms of degree and clustering coefficient. Our insights and empirical testing of previous models as well as the new model are novel contributions to the literature.

cs.SI↗