18 papers · 1 filter
Transposition Cayley Spectra and Subgraph Counts
Boris Shapiro
Let be a finite simple graph on the vertex set , and let be the Cayley graph of the symmetric group generated by the transpositions corresponding to the edges…
Real polynomials with given multiplicities of real roots: Complete conjectural description of homology
Boris Shapiro
Following \cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots. A computer-assis…
Generic -power ideals, fat points, and symbolic powers
Boris Shapiro
Problem F of \cite{FLOS18} asks whether an ideal generated by generic -power forms, where is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\big[(1…
Algebraic defect and positive mother body measures
Boris Shapiro
Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ , its irreducible equation , and a compact co…
Van Vleck spectra of high-order Heun operators:\ finite-band universality and exterior asymptotics
Boris Shapiro
We study high-order analogues of the classical Heun operator of Fuchs index one, \[ \dq=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}, \qquad °Q_i\le i+1, \qquad °Q_k=k+1. \] For a fixed deg…
Frenet turns
Boris Shapiro
We discuss a problem posed by A.~Agrachev asking how many times a usual circle in should be traversed to admit a deformation by curves with nowhere degenerating Frene…
Algebraicity of exterior Cauchy transforms of algebraic ovals: a homological formulation
Ch. Hagg, B. Shapiro
Let $Ω\subset\C$ be a bounded domain whose boundary is an oval of a real algebraic curve. We study when the exterior Cauchy transform \[ \ct_Ω(z)=\frac1π\int_Ω\frac{dA(ζ)}{z-ζ} \]…
Complex spectrum of the partial theta function
Boris Shapiro
We study the complex spectrum of the partial theta function \[ Θ(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1, \] where a spectral value is a parameter for which \(Θ(q,\cd…
Inflection curves of rational vector fields
Boris Shapiro, Guillaume Tahar
We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z…
Combinatorics of Inflection Points of Plane Curve Shadows
Boris Shapiro
We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jorda…
Canonical forms and moment-generating functions of plane polypols
Boris Shapiro
We study two closely related objects associated with plane domains bounded by rational algebraic arcs: canonical forms in the sense of positive geometry and normalized moment-gener…
Projections of convex polytopes to a line and higher univariate Prony systems
Boris Shapiro
Motivated by the inverse moment problem for convex polytopes, we study the pushforward to a line of the Lebesgue measure restricted to a convex -polytope. Such pushforwards are…
Plane rectifiable curves: old and new
Boris Shapiro, Guillaume Tahar
In this note we recall the classical notion of an algebraically rectifiable plane curve going back to J. A. Serret, E. Laguerre and G. Humbert. We provide new criteria of algebraic…
Discriminants of derivatives: square graphs and sums of squares
Boris Shapiro, Cordian Riener
The discriminants of the derivatives of a real-rooted polynomial are nonnegative polynomials in its roots, and we ask whether this nonnegativity can be certified algebraically in t…
The (n-2,2)-Spectrum of a Graph
Boris Shapiro
We study a representation-theoretic refinement of the ordinary Laplacian spectrum of a graph. Given a graph on vertices, one may associate to it the element \[ X_G=\sum_{ij…
Reducibility of spectral curves of finite Jacobi pencils
B. Shapiro
We consider finite pencils of Jacobi matrices \[ J_n(w)=A+wB, \] where is diagonal and is tridiagonal with zero diagonal. The spectral curve is the affine plane curve \[ χ_…
Ribbon graphs and meromorphic functions
B. Shapiro
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…
Confocal families of plane algebraic curves
Ragni Piene, Boris Shapiro
We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformat…