arXiv subjects

Christian Hägg

Publications and source records attributed to Christian Hägg.

10 recordsLinked to original sources

arXiv2026

The full shadow of a polynomial I. The limiting core and the transient set

The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\toα\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

math.CV↗

arXiv2026

A Bézout domain that is not an elementary divisor domain

We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.

math.AC↗

arXiv2026

An Ellipse Criterion for Exact Nonuniqueness in the Planar Interior Radon Problem

We characterize exact nonuniqueness in the planar interior Radon problem for arbitrary pairs of open convex sets. There exists a nonzero smooth function compactly supported in the first set whose integral over every line meeting the second set vanishes if and only if an ellipse contains the closure of the second set and is compactly contained in the first set. The same criterion holds for a nonzero $L^1$ function whose essential support is compactly contained in the first set and whose Radon transform vanishes almost everywhere on those lines. For concentric open squares, the criterion gives the sharp threshold $1/\sqrt2$ for the inner-to-outer half-side ratio. This yields counterexamples to Conjecture 1.2 of Boman (2021) and to Theorem 40.1 of Boman (2025).

math.CA↗

arXiv2026

Zero asymptotics for successive derivatives of hyperexponential functions with finite essential singularities

Pólya's shire theorem identifies the final set of zeros of successive derivatives of an arbitrary meromorphic function with at least one pole with the Voronoi diagram of its finite poles. We prove a fixed-scale zero-counting law for hyperexponential functions $f=(P/Q)\exp(S/T)$, allowing ordinary poles and finite essential singularities of arbitrary order and position, thus extending Pólya's picture beyond the rational, polynomial-exponential, and one-dimensional finite-essential-singularity settings. After the forced singular factors are removed from the numerator of $f^{(n)}$, the normalized zero-counting measures converge in the original $z$-plane to the classical Voronoi edge measure generated by all finite singular sites, augmented by explicitly weighted atoms at the finite essential singularities, which thereby enter Pólya's picture both as Voronoi sites and as sources of linear-size zero clusters. If $S/T$ has a nonconstant polynomial part, the complementary mass escapes to infinity. We determine the microscopic laws of these clusters, obtaining the reciprocal Marchenko--Pastur law for simple poles of $S/T$ and higher-order multiple-Laguerre, equivalently Laguerre Muttalib--Borodin, limits for higher-order poles. Finally, inside essential Voronoi cells we identify the first sublinear zero layer, including its Stokes geometry, densities, and final-set consequences away from transition loci.

math.CV↗

arXiv2026

Voronoi limit measures for iterates of constant-coefficient differential operators on rational functions with simple poles

Bøgvad and Hägg proved that for a rational function with simple poles, the zeros of successive derivatives accumulate on the Voronoi diagram of the pole set, and the normalized zero-counting measures converge to a canonical probability measure supported on this diagram. We extend this result from pure derivatives to iterates of an arbitrary monic constant-coefficient differential operator. Let $h(z)=A(z)/B(z)$ be a reduced rational function, where $B$ is monic of degree $b\ge2$ with distinct zeros $S=\{z_1,\dots,z_b\}$, and let $P(D)=\sum_{j=0}^m c_jD^j$ be a monic constant-coefficient differential operator of order $m\ge1$. After clearing denominators, we can write $P(D)^n(h)=\widetilde A_n/B^{mn+1}$ and study the zeros of the numerator polynomials $\widetilde A_n$. If $r:=\min\{j:c_j\neq0\}$, then (after passing to the proper part of $h$ when $r>0$) the associated zero-counting measures converge vaguely to $$\frac{m(b-1)}{bm-r}\,μ_S,$$ where $μ_S$ is the Bøgvad--Hägg probability measure supported on the Voronoi diagram $V_S$. In particular, the limit is a probability measure exactly when $P(D)=D^m$; otherwise a proportion $\frac{m-r}{bm-r}$ of zeros escapes to infinity (in the sense of vague convergence). When $r<m$, the unshifted logarithmic potentials diverge, but an explicit factorial renormalization yields $L^1_{\mathrm{loc}}(\mathbb C)$ convergence to a subharmonic limit with Riesz measure $\frac{m(b-1)}{bm-r}\,μ_S$. Apart from this scalar factor, the limiting measure is determined solely by the pole configuration; the coefficients of $P(D)$ affect only an additive constant in the limiting potential.

math.CV↗

arXiv2026

Sprecher Networks: A Parameter-Efficient Kolmogorov-Arnold Architecture

We introduce Sprecher Networks (SNs), a family of trainable architectures derived from David Sprecher's 1965 constructive form of the Kolmogorov-Arnold representation. Each SN block implements a "sum of shifted univariate functions" using only two shared learnable splines per block, a monotone inner spline $ϕ$ and a general outer spline $Φ$, together with a learnable shift parameter $η$ and a mixing vector $λ$ shared across all output dimensions. Stacking these blocks yields deep, compositional models; for vector-valued outputs we append an additional non-summed output block. We also propose an optional lateral mixing operator enabling intra-block communication between output channels with only $O(d_{\mathrm{out}})$ additional parameters. Owing to the vector (not matrix) mixing weights and spline sharing, SNs scale linearly in width, approximately $O(\sum_{\ell}(d_{\ell-1}+d_{\ell}+G))$ parameters for $G$ spline knots, versus $O(\sum_{\ell} d_{\ell-1}d_{\ell})$ for dense MLPs and $O(G\sum_{\ell} d_{\ell-1}d_{\ell})$ for edge-spline KANs. This linear width-scaling is particularly attractive for extremely wide, shallow models, where low depth can translate into low inference latency. Finally, we describe a sequential forward implementation that avoids materializing the $d_{\mathrm{in}}\times d_{\mathrm{out}}$ shifted-input tensor, reducing peak forward-intermediate memory from quadratic to linear in layer width, relevant for memory-constrained settings such as on-device/edge inference; we demonstrate deployability via fixed-point real-time digit classification on resource-constrained embedded device with only 4 MB RAM. We provide empirical demonstrations on supervised regression, Fashion-MNIST classification (including stable training at 25 hidden layers with residual connections and normalization), and a Poisson PINN, with controlled comparisons to MLP and KAN baselines.

cs.LG↗

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↗

arXiv2022

Introducing isodynamic points for binary forms and their ratios

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Mobius group on the set of triangles. Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d at least 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Mobius group on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

math.CV↗

arXiv2017

The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left\{\frac{\mathrm{d}^n}{\mathrm{d}z^n}\left(R(z)\exp{T(z)}\right)\right\}$. Here, $R(z)$ is a rational function with at least two poles, all of which are distinct, and $T(z)$ is a polynomial. This is an extension of a recent measure-theoretic refinement of Pólya's Shire theorem for rational functions.

math.CA↗

arXiv2017

A refinement of Pólya's method to construct Voronoi diagrams for rational functions

Given a complex polynomial $P$ with zeroes $z_1,\dotsc,z_d$, we show that the asymptotic zero-counting measure of the iterated derivatives $Q^{(n)}, \ n=1,2,\dotsc$, where $Q=R/P$ is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with $z_1,\dotsc,z_d$. This refines Pólya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in $\mathbb C^m$.

math.CA↗