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B. Shapiro

Publications and source records attributed to B. Shapiro.

At least 19 recordsLinked to original sources
arXiv2026

Finite-term recurrences in a generalized Bochner--Krall family

We classify the differential operators \(T=z^j\partial_z^j+z^m\partial_z^\ell\), where \(0\le m<\ell\) and \(1\le j<\ell\), whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing \(k=\ell-m\), such a recurrence exists if and only if \(j=1\) and \(k\mid\ell\). In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order \(\ell\). We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov--Shapiro--Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.

math-ph↗
arXiv2026

From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem

In \cite{GNS} we proved that, for every $\alpha>0$, the potential of three positive point charges has at most $12$ nondegenerate equilibrium points. We also observed that the same method would give the sharper bound $6$ if a certain auxiliary polynomial system $Q=R=0$ had at least four solutions, counted with multiplicity, in each open quadrant of the $(f,g)$-plane. Here we prove this four-solution statement. The main new ingredient is a separation argument at the unique saddle point of a separated-variable first integral. Consequently, the upper bound for three charges improves from $12$ to $6$.

math.CA↗
arXiv2026

Weak and strong $q$-analogs of the Laguerre--P\'olya class

For $0<q<1$ we compare two natural $q$-analogs of the Laguerre--P\'olya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre--P\'olya class under the normalized $q$-Borel transform \[ \Bq\left(\sum_{k\ge 0}a_k\frac{z^k}{k!}\right) =\sum_{k\ge 0}a_k\frac{q^{k(k-1)/2}(1-q)^k}{(q;q)_k}z^k . \] The second one is a zero-side class, defined as the locally uniform closure of real polynomials whose nonzero zeros are logarithmically $q$-separated on each side of the origin. We prove that the normalized $q$-Borel transform maps the classical Laguerre--P\'olya class, and its type-I subclass, into themselves. This yields a $q$-Jensen-polynomial criterion and shows that the coefficient-side class strictly contains the classical Laguerre--P\'olya class. On the zero side, we prove a genus-zero product representation. The logarithmic separation condition prevents zeros escaping to infinity from producing a residual exponential factor; consequently no nonconstant exponential factor can occur. For every $q\in(0,1)$ we obtain the strict chains \[ \qLPs\subsetneq \LP\subsetneq \qLPw, \qquad \qLPIs\subsetneq \LPI\subsetneq \qLPIw . \]

math.CA↗
arXiv2026

A generalized Stieltjes system with polynomial source

Let $Q$ be a monic polynomial of degree $M+1$. We study the algebraic system \[ \sum_{j\ne i}\frac{1}{x_i-x_j}=Q(x_i),\qquad i=1,\ldots,N, \] for pairwise distinct complex numbers $x_1,\ldots,x_N$, modulo permutations of these numbers. The case $M=0$ is, after a translation, the classical Stieltjes system for the zeros of a Hermite polynomial. We prove that, for arbitrary $Q$, the number of solutions is at most $\binom{N+M}{N}$, and that the coefficient equations for the associated monic Stieltjes polynomial have total intersection multiplicity exactly $\binom{N+M}{N}$. Consequently the bound is attained for all $Q$ in a non-empty Zariski open subset of the affine space of monic polynomials of degree $M+1$. We also describe the solutions when the coefficient of the linear term of $Q$ is large: the system splits into $M+1$ weakly coupled classical Stieltjes systems, one near each zero of $Q$.

math-ph↗
arXiv2026

Algebraicity of exterior Cauchy transforms of algebraic ovals: a homological formulation

Let $\Omega\subset\C$ be a bounded domain whose boundary is an oval of a real algebraic curve. We study when the exterior Cauchy transform \[ \ct_\Omega(z)=\frac1\pi\int_\Omega \frac{dA(\zeta)}{z-\zeta} \] is algebraic. The boundary formula identifies this transform with a Cauchy-type integral on the normalization $X$ of the relevant irreducible component of the Schwarz correspondence $P(z,w)=0$. The main point is that $X$ is fixed while only the divisor $\pi^{-1}(z)$ of moving poles varies. Thus the natural monodromy is point-pushing on a punctured fixed surface and becomes trivial on absolute homology after the moving punctures are filled; in particular, the usual Picard--Lefschetz transvection picture does not produce absolute cycles in this problem. The principal theorem is a residue criterion: if the lifted boundary is separating, i.e., if it bounds an integral two-chain on $X$ disjoint from the fixed polar divisor, then the exterior Cauchy transform is algebraic and is given by an explicit residue sum with chain multiplicities. This implies, in particular, algebraicity for every smooth oval on a rational real algebraic curve and for separating ovals in positive genus. We also record the corresponding complete-real-locus statement for dividing real curves, with the necessary affine-plane caveats. Nonseparating ovals are treated as a conjectural period problem: we formulate a period-rank test which can detect possible Abelian contributions. The examples include the ellipse, the nodal cubic logarithm, a smooth Weierstrass cubic as an elliptic-period test case, and a conditional positive-genus construction illustrating algebraic transforms beyond quadrature domains.

math.AG↗
arXiv2026

Reducibility of spectral curves of finite Jacobi pencils

We consider finite pencils of Jacobi matrices \[ J_n(w)=A+wB, \] where $A$ is diagonal and $B$ is tridiagonal with zero diagonal. The spectral curve is the affine plane curve \[ \chi_n(\lambda,w)=\det(\lambda I+J_n(w))=0 . \] The main question is to describe when this curve is reducible. We prove generic irreducibility for fixed pairwise distinct diagonal entries and discuss several elementary reducibility mechanisms. Besides disconnected Jacobi chains, constant eigenvalue branches, and reflection-symmetric components, one must also take into account reducibility caused by scalar diagonal blocks. We formulate a reducibility conjecture and record low-dimensional evidence and counterexamples to several overly optimistic classifications. A central point of the picture is a codimension-growth principle: apart from the cutting divisors $b_i=0$, genuinely connected primitive reducibility should move to higher and higher codimension as the size of the chain grows.

math.SP↗
arXiv2026

Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws

We study the asymptotic distribution of level crossings for random matrix pencils A_n+\lambda B_n in several ensembles, including complex and real i.i.d. matrices and Gaussian/Hermitian settings. We derive a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and formulate conditions under which its limit is governed by a deterministic potential. Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, we prove convergence of the empirical measure of level crossings to an explicit deterministic limit. In the complex Gaussian case these assumptions are verified (modulo a uniformity step), while in the general i.i.d. setting the results are conditional and motivated by universality theory. We further analyze the real case, showing that any limiting measure does not concentrate on the real projective line under suitable hypotheses, and discuss analogous phenomena for elliptic/Hermitian ensembles. Our results highlight the role of logarithmic energy and universality in governing spectral degeneracies of random matrix pencils.

math-ph↗
arXiv2026

Ribbon graphs and meromorphic functions

Let Y be a compact Riemann surface, phi:Y -> CP^1 a meromorphic function, and Gamma in Y a ribbon graph avoiding the critical points of phi. Then phi(Gamma) is an immersed graph in CP^1. Conversely, given an immersion im:Theta to bCP^1 of an abstract multigraph Theta without vertices of valence 1 or 2, we describe a construction of a compact Riemann surface Y and a meromorphic function phi_{im}:Y in CP^1 such that phi_{im}(Gamma)=im(Theta). We investigate the relation between the topology of Y and the combinatorics of Gamma. In particular, for a surface of genus g we construct spanning ribbon graphs whose underlying abstract graphs have arbitrary prescribed graph genus g' smaller or equal g, including the planar case. As a consequence, the number of self-intersections of \phi(Gamma) cannot, in general, be controlled solely by the genus of Y. We establish general lower bounds for the number of self-intersections and formulate several open problems, with emphasis on planar ribbon graphs.

math.AG↗
arXiv2021

Universal statistics of waves in a random time-varying medium

We study the propagation of waves in a medium in which the wave velocity fluctuates randomly in time. We prove that at long times, the statistical distribution of the wave energy is log-normal, with the average energy growing exponentially. For weak disorder, another regime preexists at shorter times, in which the energy follows a negative exponential distribution, with an average value growing linearly with time. The theory is in perfect agreement with numerical simulations, and applies to different kinds of waves. The existence of such universal statistics bridges the fields of wave propagation in time-disordered and space-disordered media.

cond-mat.dis-nn↗
arXiv2017

Activation of Microwave Fields in a Spin-Torque Nano-Oscillator by Neuronal Action Potentials

Action potentials are the basic unit of information in the nervous system and their reliable detection and decoding holds the key to understanding how the brain generates complex thought and behavior. Transducing these signals into microwave field oscillations can enable wireless sensors that report on brain activity through magnetic induction. In the present work we demonstrate that action potentials from crayfish lateral giant neuron can trigger microwave oscillations in spin-torque nano-oscillators. These nanoscale devices take as input small currents and convert them to microwave current oscillations that can wirelessly broadcast neuronal activity, opening up the possibility for compact neuro-sensors. We show that action potentials activate microwave oscillations in spin-torque nano-oscillators with an amplitude that follows the action potential signal, demonstrating that the device has both the sensitivity and temporal resolution to respond to action potentials from a single neuron. The activation of magnetic oscillations by action potentials, together with the small footprint and the high frequency tunability, makes these devices promising candidates for high resolution sensing of bioelectric signals from neural tissues. These device attributes may be useful for design of high-throughput bi-directional brain-machine interfaces.

physics.app-ph↗
arXiv2016

Effect of the type I to type II Weyl semimetal topological transition on superconductivity

The influence of recently discovered topological transition between type I and type II Weyl semi-metals on superconductivity is considered. A set of Gorkov equations for weak superconductivity in Weyl semi-metal under topological phase transition is derived and solved. The critical temperature and superconducting gap both have spike in the point the transition point as function of the tilt parameter of the Dirac cone determined in turn by the material parameters like pressure. The spectrum of superconducting excitations is different in two phases: the sharp cone pinnacle is characteristic for a type I, while two parallel almost flat bands, are formed in type II. Spectral density is calculated on both sides of transition demonstrate different weight of the bands. The superconductivity thus can be used as a clear indicator for the topological transformation. Results are discussed in the light of recent experiments.

cond-mat.supr-con↗
arXiv2016

"K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs

In this paper we study a filtered "K-theoretical" analog of a graded algebra associated to any loopless graph G which was introduced in \cite{PS}. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above "K-theoretical" analog.

math.CO↗
arXiv2016

Level Crossing in Random Matrices: I. Random perturbation of a fixed matrix

We consider level crossing in a matrix family $H=H_0+\lambda V$ where $H_0$ is a fixed $N\times N$ matrix and $V$ belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of $\lambda$, for which we obtain a number of exact, asymptotic and approximate formulas.

math-ph↗
arXiv2015

A tropical analog of Descartes' rule of signs

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_d of positive numbers such that, for any real polynomial P of degree d, the number of its real roots is less than or equal to the number of the so-called essential tropical roots of the polynomial obtained from P by multiplication of its coefficients by a_0, a_1,... a_d respectively. In particular, for any real univariate polynomial P of degree d with non-vanishing constant term, we conjecture that one can take a_k = e^{-k^2}, k = 0, ... , d. The latter claim can be thought of as a tropical generalization of Descartes's rule of signs. We settle this conjecture up to degree 4 as well as a weaker statement for arbitrary real polynomials. Additionally we describe an application of the latter conjecture to the classical Karlin problem on zero-diminishing sequences.

math.CA↗
arXiv2015

On asymptotic Gauss-Lucas theorem

In this note we extend the Gauss-Lucas theorem on the zeros of the derivative of a univariate polynomial to the case of sequences of univariate polynomials whose almost all zeros lie in a given convex bounded domain in C.

math.CA↗
arXiv2014

On spectral asymptotic of quasi-exactly solvable quartic

Motivated by the earlier results, we study theoretically and numerically the asymptotics and the monodromy of the quasi-exactly solvable part of the spectrum of the quasi-exactly solvable quartic introduced by C.~M.~Bender and S.~Boettcher. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the configuration of the branching points of the latter quartic with the asymptotic shape of zeros of the Yablonskii-Vorob'ev polynomials recently described and present its (conjectural) alternative description. Further we present a numerical study of the spectral monodromy for the os- cillator in question.

math-ph↗
arXiv2014

A note on planarity stratification of Hurwitz spaces

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to CP^2 and a projection of the image curve from an appropriate point p in CP^2 to the pencil of lines through p. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata.

math.AG↗