Algebraic Curves Research Articles

Explicit matrices with low coherence based on algebraic geometric codes

A key element in the performance of a compressed sensing (CS) setup is the so called sensing matrix. It is known that the success of CS-based methods strongly rely on the properties of the employed sensing matrix. Random matrices are the widely-adopted choice due to their order-optimal performance and flexibility in size. In real world applications, however, random structures are rarely feasible. For this reason, the deterministic design of sensing matrices has been an ongoing research topic. In this paper, we introduce new classes of deterministic complex-valued sensing matrices based on algebraic curves. In particular, we design a number of algebraic-geometric codes with large minimum distances specifically for the construction of sensing matrices. Our approach is to find maximal curves in the Galois field Fpm‾, and transform them into Fp codes by a trace map. Invoking the Riemann-Roch theorem, we demonstrate that the resulting code has a large minimum distance compared to its length, which leads to a sensing matrix with small coherence value. For general m×n matrices, the Welch bound (≈1m) sets a universal lower-bound on the coherence value, and the bound is achievable only when n≤m2. In our designs, we are able to construct m×n matrices with n ranging from around 8m to values larger than O(m2) by tuning the parameters. Meanwhile, the coherence of the designed matrices differ from the Welch bound by only an O(logm) factor. Simulation results indicate that the performance of our matrices in recovering sparse vectors from compressed measurements is superior or equivalent to Gaussian random matrices.

Rational points close to non‐singular algebraic curves

AbstractWe study the density of solutions to Diophantine inequalities involving non‐singular ternary forms, or equivalently, the density of rational points close to non‐singular plane algebraic curves.

Burchnall–Chaundy polynomials for matrix ODOs and Picard–Vessiot Theory

Burchnall and Chaundy showed that if two ordinary differential operators (ODOs) P, Q with analytic coefficients commute then there exists a polynomial f(λ,μ) with complex coefficients such that f(P,Q)=0, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Our matrices have entries in a differential field K, whose field of constants C is algebraically closed and of zero characteristic. We restrict to the case of order one operators P, with invertible leading coefficient. We define a new differential elimination tool, the matrix differential resultant. We use it to compute the BC-polynomial f of a pair of commuting MODOs, and we also prove that it has constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem PY=λY,QY=μY to have a solution. Techniques from differential algebra and Picard–Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs C[P,Q] and a finite product of rings of irreducible algebraic curves.

A new class of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{gl}_r)$-algebras

Quantum rr-Airy structures can be constructed as modules of \mathcal{W}(\mathfrak{gl}_r)𝒲(𝔤𝔩r)-algebras via restriction of twisted modules for the underlying Heisenberg algebra. In this paper we classify all such higher quantum Airy structures that arise from modules twisted by automorphisms of the Cartan subalgebra consisting of products of disjoint cycles of the same length. An interesting feature of these higher quantum Airy structures is that the dilaton shifts must be chosen carefully to satisfy a matrix invertibility condition, with a natural choice being roots of unity. We explore how these higher quantum Airy structures may provide a definition of the Chekhov, Eynard, and Orantin topological recursion for reducible algebraic spectral curves. We also study under which conditions quantum rr-Airy structures that come from modules twisted by arbitrary automorphisms can be extended to new quantum (r+1)(r+1)-Airy structures by appending a trivial one-cycle to the twist without changing the dilaton shifts.

Globally certified [formula omitted] approximation of planar algebraic curves

Topology analysis and geometric feature determination for curves and surfaces are basic problems in geometric modeling and computer aided design. When the curves and surfaces are given in implicit expressions, the problems become more difficult. This paper presents a complete algorithm framework for the problem of certified G1 approximation of a planar algebraic curve under global error control. Compared to other work on topology and geometric analysis of algebraic curves, this method presents a new topology analysis as well as a geometric information computation. Moreover, this method uses generalized asymptotes as the approximation of the unbounded part, which makes it achieve global approximation with given precision.

Galois $$E_6$$-Bundles over a Hyperelliptic Algebraic Curve

Galois $$E_6$$-Bundles over a Hyperelliptic Algebraic Curve

Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

Incidence varieties in the projectivized kth Hodge bundle over curves with rational tails

Over the moduli space of pointed smooth algebraic curves, the projectivized [Formula: see text]th Hodge bundle is the space of [Formula: see text]-canonical divisors. The incidence loci are defined by requiring the [Formula: see text]-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized [Formula: see text]th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.

Crossing the Transcendental Divide: From Translation Surfaces to Algebraic Curves

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins–Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.

Poincaré–Reeb graphs of real algebraic domains

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincaré–Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré–Reeb graph.

On the numerical range of Kac-Sylvester matrices

In this paper, the boundary generating curves and the numerical range of Kac-Sylvester matrices up to the order $9$ are characterized. Based on the obtained results and on several computational experiments performed with the Mathematica and MatLab programs, we conjecture that the found types of algebraic curves, namely ellipses and ovals, will appear for an arbitrary order.

The Tropical Manin–Mumford Conjecture

Abstract In analogy with the Manin–Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel–Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus ${g\geq 2}$, which are biconnected and have edge lengths that are “sufficiently irrational” in a precise sense. Under these assumptions, the number of torsion points is bounded by $3g-3$. Next, we study bounds on the number of torsion points in the image of higher-degree Abel–Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This motivates the definition of the “independent girth” of a graph, a number that is a sharp upper bound for $d$ such that the higher-degree Manin–Mumford property holds.

The Newton–Puiseux Algorithm and Triple Points for Plane Curves

The paper is an introduction to the use of the classical Newton–Puiseux procedure, oriented towards an algorithmic description of it. This procedure allows to obtain polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and is almost self-contained. We illustrate the use of the algorithm first in a completely worked out example of a curve with a point of multiplicity 6, and secondly, in the study of triple points on reduced plane curves.

The truncated moment problem on curves y = q(x) and yx ℓ = 1

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form y = q ( x ) , q ( x ) ∈ R [ x ] , deg ⁡ q ≥ 3 and y x ℓ = 1 , ℓ ∈ N ∖ { 1 } . For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. http://doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022.126563.], and improve Fialkow's bound to deg ⁡ q − 1 (resp. ℓ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are kdeg ⁡ q (resp. k ( ℓ + 1 ) ) for curves y = q ( x ) (resp. y x ℓ = 1 ) for even degree sequences, while for odd ones they are kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. k ( ℓ + 1 ) − ⌊ ℓ 2 ⌋ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves y = x ℓ , ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve y = x 2 .

Arithmetic inflection formulae for linear series on hyperelliptic curves

AbstractOver the complex numbers, Plücker's formula computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in ‐homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.

Reconstruction anabélienne du squelette des courbes analytiques

In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by Andr\'e, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language of semi-graphs of anabelio\ids and tempero\ids. Our work consists in associating a graph of anabelio\ids to a Berkovich curve equipped with a minimal triangulation and in adapting the results of Mochizuki in order to recover the analytic skeleton of the curve. The novelty of this anabelian result in Berkovich geometry is that the curves we are interested in are not supposed anymore to be of algebraic nature. We show for example that the famous Drinfeld half-plane is an analytically anabelian curve.

Orthogonal polynomials on a class of planar algebraic curves

AbstractWe construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form in where and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree are computed via the Lanczos algorithm in operations.

On one-dimensional local rings and Berger’s conjecture

AbstractLet k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module ${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say $\mathfrak{h}$, we show that A is regular whenever the ring $A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.

Calculation of a Strong Resonance Condition in a Hamiltonian System

A method for symbolic computation of a condition of existence of a third- and fourth-order resonance for investigations of formal stability of an equilibrium state of a multiparameter Hamiltonian system with three degrees of freedom in the case of general position is proposed. This condition is formulated in the form of zeros of a quasi-homogeneous polynomial of the coefficients of the characteristic polynomial of the linear part of the Hamiltonian system. Computer algebra (Gröbner bases of elimination ideals) and power geometry (power transformations) are used to represent this condition for various resonance vectors in the form of rational algebraic curves. Given a linear approximation of the characteristic polynomial in the space of its coefficients, these curves are used to obtain a description of a partition of the stability domain into parts in which there are no strong resonances. An example of a description of resonance sets for a two-parameter pendulum-type system is given. All computations are carried out in the computer algebra system Maple.

A Survey of Riemann-Roch-type Theorems

The Riemann-Roch theorem is a classical result which forms a beautiful algebraic connection between complex analysis on a compact Riemann surface and a global topological property of that surface (the genus). We present a survey of the theorem and its many variants and generalizations. We also provide an alternative elementary proof of the Riemann-Roch formula for algebraic curves and outline some applications of the formula.