arXiv subjects

Boris Shapiro

Publications and source records attributed to Boris Shapiro.

At least 19 recordsLinked to original sources

arXiv2026

The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry

We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graph algebras and defined by a deletion-loopy-contraction recursion, in which the contracted edge becomes a loop. We show that L_G contains the Tutte polynomial and admits a similar spanning-forest activity expansion. It also determines Stanley's chromatic symmetric function, the degree sequence, the induced edge count profile, and the independence polynomial for loopless graphs, and the clique and matching polynomials for simple graphs. Separating the size and the external activity of each forest component leads to a refined loopy polynomial, which we show to be equivalent to the extended U-polynomial of Noble and Welsh and to the extended polychromate. Different specializations of this common refinement give the ordinary U-polynomial, Tutte's universal V-function, and Stanley's Tutte symmetric function, placing these invariants into a single framework. We conjecture that L_G and the U-polynomial have the same distinguishing power on simple graphs, and verify this for all graphs on at most 11 vertices. Simplicity is essential: we found two loopless multigraphs with equal U-polynomials but distinct loopy polynomials. They also have distinct extended U-polynomials, so the ordinary U-polynomial does not determine the extended one on loopless multigraphs. This solves an open problem by Merino and Noble. For the score polytope P_G of the external bizonotopal algebra, loopy deletion-contraction lifts from the lattice-point enumerator to the polytope itself. This gives forest-indexed geometric parking complexes whose lattice points partition those of P_G, and which are piecewise-linearly parametrized by products of intervals whose lengths are the component weights in the forest expansion of L_G.

math.CO↗

arXiv2026

Real polynomials with given multiplicities of real roots: Complete conjectural description of homology

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-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $\omega=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$. We relate that count to the rational signed weight enumerator $F_\omega(t)=\sum_{\eta\preceq\omega}(-1)^{\elln(\eta)}t^{|\eta|}$. For the counterexample, $F_\omega(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded. We prove a number of results and formulate a complete conjecture describing the above homology.

math.AT↗

arXiv2026

Algebraic defect and positive mother body measures

Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ $f$, its irreducible equation $P(z,w)=0$, and a compact convex set $K$ the functional \[ \mathfrak D_{P,K}(\mu)= \int_K\left|P\bigl(z,\mathcal C_\mu(z)\bigr)\right|^{1/d}\,dA(z), \qquad d=\text{deg}_wP, \] on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of $\mathfrak D_{P,K}(\mu)$ and attainment of its minimum, characterize zero defect, and obtain the estimate \[ \mu(D(a,r))\le C_1r+C_2\mathfrak D_{P,K}(\mu)/r \] away from the zero set of the leading coefficient of $P$, where $D(a,r)$ is the disk of radius $r$ centered at $a$. We establish a dual formula for the defect in the rational case and construct positive algebraic Cauchy transforms by Herglotz theory, Fuss--Catalan and Raney laws, positive sums, polynomial pushforwards, and branch graphs. The passage from zero defect to a mother body is made under the planar-null hypothesis of the support.

math.CA↗

arXiv2026

Van Vleck spectra of high-order Heun operators:\ finite-band universality and exterior asymptotics

We study high-order analogues of the classical Heun operator of Fuchs index one, \[ \dq=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}, \qquad \deg Q_i\le i+1, \qquad \deg Q_k=k+1. \] For a fixed degree $n$ we consider the linear Van Vleck polynomials $V$ for which $\dq+V$ has a polynomial solution of degree $n$, and we form the spectral polynomial $Sp_n$ whose zeros are the zeros of these Van Vleck polynomials. The main result is a finite-band determinant representation and the resulting universality theorem: after normalization, all fixed power sums of the zeros of $Sp_n$ have limits given by explicit constant-term formulae depending only on the leading coefficient $Q_k$. The lower coefficients of $\dq$ enter only lower order correction terms. Combining this with the localization theorem for Van Vleck roots, we strengthen the usual germ-at-infinity conclusion to locally uniform convergence of the normalized Cauchy transforms and logarithmic potentials on the whole exterior of the convex hull of the zeros of $Q_k$. We also prove a determinacy criterion: if the spectral roots are asymptotically confined to a compact set with empty interior and connected complement, then the finite-band moments determine the actual weak limit. In particular, when the zeros of $Q_k$ are collinear the root-counting measures of $Sp_n$ converge weakly to a probability measure supported on the corresponding segment; this limit is independent of all lower coefficients of $\dq$. Finally, we prove holonomicity of the exterior Cauchy transform and derive Picard--Fuchs equations for the WKB periods, with an explicit third-order equation in the first non-classical case $k=3$. The paper ends with a precise mother-body conjecture for the genuinely complex case, clearly separated from the unconditional results.

math-ph↗

arXiv2026

Frenet turns

We discuss a problem posed by A.~Agrachev asking how many times a usual circle in $\mathbb R^n$ should be traversed to admit a deformation by curves with nowhere degenerating Frenet frame. It turns out that the answer depends on a specific topology which we consider. For the literal $C^n$ curve topology, the least number of turns of a plane circle admitting arbitrarily small nondegenerate perturbations is \[ k(2)=1,\qquad k(3)=2,\qquad k(n)=1\quad(n\ge4). \] This jet-level problem is different from the original Frenet-control problem by Agrachev. We show that in the literal interpretation of Agrachev's problem one has a simple spherical Fenchel obstruction in all dimensions $n\ge4$. To retain a nontrivial turn-counting problem, we introduce decorated turn data. In $\R^4$ the datum is a pair $(p,q)$ recording tangent-plane and normal-plane turns; we prove that every nonresonant pair $(p,q)$ with $p,q>0$, $p\ne q$, is accessible by small positive constant Frenet controls. In even dimension $2r$ the analogous datum is a vector $(p_1,\ldots,p_r)$, and every vector with pairwise distinct positive entries is accessible by constant controls. Odd dimensions require genuinely time-dependent openings since constant controls cannot close the base curve.

math.DG↗

arXiv2026

Complex spectrum of the partial theta function

We study the complex spectrum of the partial theta function \[ \Theta(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1, \] where a spectral value is a parameter for which $\Theta(q,\cdot)$ has a multiple zero. Since the function is defined here only for $|q|<1$, all spectral values are strictly inside the unit disk; boundary points on $|q|=1$ occur only as accumulation points of the spectrum. The paper combines two complementary points of view. Near the unit circle we prove that every point of $|q|=1$ is an accumulation point of the spectrum; the proof uses explicit spectral factors of truncations, the Jacobi triple product, and a boundary-window lifting argument near roots of unity. Inside a fixed subdisk, illustrated for $|q|\leq 0.8$, the true spectrum is locally finite and must be separated carefully from the much larger branch loci of truncations and Jensen polynomials. We give a truncation-seeded Newton procedure which produces a discrete list of candidate spectral values, explain the caustic/escaping-root mechanism in finite approximants, and record numerical monodromy experiments using a radial convention: for a spectral point $q_*$, roots are labelled at the point $0.1q_*/|q_*|$ on the small circle and then continued along the straight radial segment to $q_*$. This convention gives a coherent set of collision labels in the disk, treats negative real spectral values from the base point $-0.1$, and leads to a preliminary rational-direction heuristic for radial monodromy.

math.CA↗

arXiv2026

Inflection curves of rational vector fields

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)}{P(z)} $ we define its affine regular inflection locus by $ \{z\in \mathbb{C}: R(z)\ne0,\ P(z)\neq 0,\ \text{Im} R'(z)=0\} $ and its algebraic inflection curve by the closure of this locus, equivalently by $ \mathfrak{I}_R=(R')^{-1}(\mathbb{RP}^1). $ We prove an explicit defining equation, degree bounds, local normal forms near poles, the asymptotic directions at infinity, and a maximum-principle obstruction excluding compact components without poles. We also explain that these curves are precisely the real dessins associated with exact rational differentials, i.e. rational differentials with zero residues. Finally, we give a reducibility criterion for the complexification, prove a generic irreducibility statement in the usual separated-variable sense, and classify the exact dessins of degree at most two. The degree-three problem is reduced to three explicit normal forms.

math.DS↗

arXiv2026

Combinatorics of Inflection Points of Plane Curve Shadows

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 Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree--necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit $\mathbb Z_2$ holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This ``parallel-tangent load'' is controlled by the same inflection folds but is not determined by their number alone.

math.GT↗

arXiv2026

Discriminants of derivatives and symmetric difference polynomials

Let $P$ be a monic polynomial of degree $n$ with roots $x_1,\ldots,x_n$. We study the discriminants of the derivatives $P^{(k)}$ as symmetric translation-invariant polynomials in the original roots. Alexandersson and Shapiro conjectured that every such discriminant belongs to the cone generated by symmetrized graph monomials with even edge multiplicities. We obtain a sharp positive/negative picture in the first terminal cases. For the terminal cubic family $k=n-3$ we prove the conjecture for every $n\ge3$, and for $n\ge5$ obtain the three-graph formula conjectured in \cite[Example~3]{AS}. For the terminal quartic family we show, by exact finite computation, that $\disc(P^{(n-4)})$ belongs to the square-graph cone if and only if $4\le n\le22$. Thus the general square-graph cone conjecture fails already at $(n,k)=(23,19)$ and fails for every $k=n-4$ with $n\ge23$. The negative result is certified by an explicit linear functional which is nonnegative on every degree-$12$ square-graph generator and strictly negative on the quartic discriminant. We also record central-moment formulas, the subset-average and finite Appell structure of normalized terminal polynomials, and the explicit quintic member.

math.CA↗

arXiv2026

The (n-2,2)-Spectrum of a Graph

We study a representation-theoretic refinement of the ordinary Laplacian spectrum of a graph. Given a graph $G$ on $n$ vertices, one may associate to it the element \[ X_G=\sum_{ij\in E(G)} (ij)\in \C[S_n]. \] The action of $X_G$ in irreducible representations of $S_n$ produces spectral invariants of graphs. The standard representation $(n-1,1)$ recovers the ordinary graph Laplacian spectrum, up to the elementary affine change $X_G=mI-L_G$, where $m=|E(G)|$. The next component, $(n-2,2)$, gives the first representation-theoretic correction. We give an explicit edge-space model for this component, derive a concrete coordinate formula for the induced operator, give a conceptual formula for all trace moments, specialize it to trees as universal linear combinations of support-forest counts, and then compute the first three moments explicitly. The third moment is expressed in terms of three-edge subgraph counts. We also introduce a weighted trace polynomial and prove that this weighted refinement already reconstructs every tree from the second moment, except for a single exceptional value of $n$ where the fourth moment suffices. Finally we discuss the relation with the invariant-theoretic approach of Thi\'ery \cite{Thiery} and formulate a more explicit support-forest-profile conjecture for the unweighted graph isomorphism problem for trees.

math.CO↗

arXiv2026

Projections of convex polytopes to a line and higher univariate Prony systems

Motivated by the inverse moment problem for convex polytopes, we study the pushforward to a line of the Lebesgue measure restricted to a convex $d$-polytope. Such pushforwards are spline densities of degree $d-1$, and their moments lead naturally to a family of ``higher'' univariate Prony systems, with the classical Prony system recovered when $d=0$. We describe the corresponding fixed-knot spline cone, give an explicit amplitude recovery criterion, record the rational generating function and recurrence satisfied by the normalized moments, and identify the directional moment variety with the Hankel determinantal variety appearing in the theory of moment varieties of measures on polytopes.

math.CA↗

arXiv2026

Canonical forms and moment-generating functions of plane polypols

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-generating functions, or Fantappie transforms. For polygons these objects are related by polarity: the normalized Fantappie transform of a polygon is the canonical form of the polar polygon. For genuinely curved polypols the same dual-geometric mechanism survives, but the transform is no longer a rational logarithmic canonical form; rather, it is a holonomic, generally branched period whose singularities are controlled by vertex hyperplanes and by the projective dual curves of the nonlinear boundary components. We give explicit examples, including sectors and half-disks, and explain how harmonic moment generating functions arise as one-dimensional restrictions of the same Fantappi`e transform.

math.AG↗

arXiv2026

Plane rectifiable curves: old and new

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 rectifiability, relate this notion to quadratic differentials, and generalize it to differentials of higher order.

math.AG↗

arXiv2026

Confocal families of plane algebraic curves

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 deformation theory.

math.AG↗

arXiv2026

Structured perturbations of tridiagonal twisted Toeplitz matrices

Twisted Toeplitz matrices constitute a generalization of Toeplitz matrices in the sense that the entries on each diagonal no longer need to be constant, but are given by the values of a continuous function on a partition of $[0,1]$. We study the limiting statistical distribution of the eigenvalues of matrices of the form $R_n(a) = T_n(a) + \sigma_n X_n$, where $T_n(a)$ is a sequence of non-Hermitian tridiagonal twisted Toeplitz matrices, $X_n$ is a sequence of tridiagonal random matrices whose entries have mean $0$ and finite variance, and $\sigma_n\to0$. The limiting distribution turns out to be a two-dimensional measure which is in general different from the push-forward of the Lebesgue measure by the symbol. We also explain how the results could extend to banded non-Hermitian twisted Toeplitz matrices.

math.PR↗

arXiv2026

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

P\'olya'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 P\'olya'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 P\'olya'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

Carrier Drift Modulation and the Hyperbolic Time Crystals

We introduce the Carrier Drift Modulation - a new mechanism for creating temporal boundaries and enabling photonic time crystals. This approach opens a direct route to hyperbolic temporal metamaterials and, in particular, hyperbolic time crystals. We demonstrate that the very process responsible for time crystal formation can simultaneously compensate for intrinsic material losses in the supporting medium - overcoming one of the central challenges in nanophotonics. The realization of truly lossless hyperbolic media, long considered as one of the key challenges of nanophotonics, unlocks new possibilities for subwavelength light focusing, strong-field physics, and novel regimes of light-matter interaction. Crucially, the proposed approach can be implemented using existing materials and readily available light sources, making it both practical and transformative.

physics.optics↗

arXiv2025

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 $\phi$ and a general outer spline $\Phi$, together with a learnable shift parameter $\eta$ and a mixing vector $\lambda$ 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↗