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Rikard Bøgvad

Publications and source records attributed to Rikard Bøgvad.

10 recordsLinked to original sources
arXiv2026

The translation geometry of Pólya's shires

In his shire theorem, G. Pólya proves that the zeros of iterated derivatives of a meromorphic function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. By recasting the local arguments of Pólya into the language of translation surfaces, we prove its generalisation describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator $T_ω: f \mapsto \frac{df}ω$ where $ω$ is a given meromorphic $1$-form. The accumulation set of these zeros is the union of edges of a generalised Voronoi diagram defined by the initial function $f$ together with the singular flat metric on the Riemann surface induced by $ω$. This result provides the ground for a novel approach to the problem of finding a flat geometric presentation of a translation surface initially defined in terms of algebraic or complex-analytic data.

math.GT↗
arXiv2023

Rodrigues' descendants of a polynomial and Boutroux curves

Motivated by the classical Rodrigues' formula, we study the root asymptotic of the polynomial sequence $$R_{[αn],n,P}(z)=\frac{d^{[αn]}P^n(z)}{dz^{[αn]}}, n= 0,1,\dots$$ where ${P(z)}$ is a fixed univariate polynomial, $α$ is a fixed positive number smaller than deg $P$, and $[αn]$ stands for the integer part of $αn$. Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced by Bertola. As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by $\{R_{[αn],n,P}(z)\}$ as well as higher derivatives of powers of more general functions.

math.CA↗
arXiv2018

Length and decomposition of the cohomology of the complement to a hyperplane arrangement

Let $\mathcal A$ be a hyperplane arrangement in $\mathbb C^n$. We show that the number of decomposition factors as a perverse sheaf of the direct image $Rj_*\mathbb C_U $ of the constant sheaf on the complement $U$ to the arrangement is given by the Poincaré polynomial of the arrangement. Furthermore we describe the composition factors of $Rj_*\mathbb C_U $ as certain local cohomology sheaves and give their multiplicity.

math.AG↗
arXiv2017

Decomposition of Perverse Sheaves on Plane Line Arrangements

On the complement $X= {\mathbb C}^2 - \bigcup_{i=1}^n L_i$ to a central plane line arrangement $\bigcup_{i=1}^n L_i \subset {\mathbb C}^2$, a locally constant sheaf of complex vector spaces $\mathcal L_a$ is associated to any multi-index $a \in {\mathbb C}^n$. Using the description of MacPherson and Vilonen of the category of perverse sheaves (\cite{MV2} and \cite {MV3}) we obtain a criterion for the irreducibility and number of decomposition factors of the direct image $Rj_* \mathcal L_a$ as a perverse sheaf, where $j: X \rightarrow {\mathbb C}^2$ is the canonical inclusion.

math.AT↗
arXiv2014

Zeros of a certain class of Gauss hypergeometric polynomials

In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[ \begin{array}{c} -n, αn+1\\ αn+2 \end{array} ; \begin{array}{cc} z \end{array}\right]$$ cluster on a certain curve defined as a part of a level curve of an explicit harmonic function. This generalizes work by Boggs, Driver, Duren et. al, to a complex parameter $α$.

math.CV↗
arXiv2013

Asymptotic distribution of zeros of a certain class of hypergeometric polynomials

We study the weak asymptotic behavior of the zeros of a family of a certain class of (generalized) hypergeometric polynomials, using the associated hypergeometric differential equation, as the parameters go to infinity. We describe the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation. In a certain degenerate case we make a precise conjecture, based and generalizing work by Duren, Driver et. al, and present experimental evidence for it.

math.CV↗
arXiv2010

Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\la}=\sum_{i=0}^k Q_{i}(z)\la^{k-i}\frac {d^i}{dz^i}$, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $\la$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $\la_{n,j}$, $1\le j\le k$, of the spectral parameter $\la$ such that the operator $T_{\la}$ has a polynomial eigenfunction $p_{n,j}(z)$ of degree $n$. These eigenfunctions split into $k$ different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits $Ψ_{j}(z)=\lim_{n\to\infty} \frac{p_{n,j}'(z)}{\la_{n,j}p_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation $\sum_{i=0}^k Q_{i}(z) Ψ_{j}^i(z)=0$ almost everywhere in $\bCP$. As a consequence we obtain a class of algebraic functions possessing a branch near $\infty\in \bCP$ which is representable as the Cauchy transform of a compactly supported probability measure.

math.CA↗
arXiv2009

Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures

We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.

math.CV↗
arXiv2009

Piecewise harmonic subharmonic functions and positive Cauchy transforms

We give a local characterization of the class of functions having positive distributional derivative with respect to $\bar{z}$ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild non-degeneracy assumptions. As a consequence, we give conditions that guarantee that any subharmonic piecewise harmonic function coincides locally with the maximum of finitely many harmonic functions and we describe the topology of their level curves. These results are valid in a quite general setting as they assume no {\em à priori} conditions on the differentiable structure of the support of the associated Riesz measures. We also discuss applications to positive Cauchy transforms and we consider several examples and related problems.

math.CV↗