Theory Of Algebraic Curves Research Articles

Algebro-geometric constructions of semi-discrete Chen–Lee–Liu equations

Resorting to the finite-order expansion of the Lax matrix, the relation between elliptic coordinates and potentials is established, from which the semi-discrete Chen–Lee–Liu equations are decomposed into solvable ordinary differential equations. Based on the theory of algebraic curves, Abel–Jacobi coordinates are introduced to straighten out the continuous flow and discrete flow, by which explicit solutions of the semi-discrete Chen–Lee–Liu equations are obtained in the Abel–Jacobi coordinates.

Decomposition of the Discrete Ablowitz–Ladik Hierarchy

The nonlinearization approach of Lax pairs is extended to the discrete Ablowitz–Ladik hierarchy. A new symplectic map and a class of new finite‐dimensional Hamiltonian systems are derived, which are further proved to be completely integrable in the Liouville sense. An algorithm to solve the discrete Ablowitz–Ladik hierarchy is proposed. Based on the theory of algebraic curves, the straightening out of various flows is exactly given through the Abel–Jacobi coordinates. As an application, explicit quasi‐periodic solutions for the discrete Ablowitz–Ladik hierarchy are obtained resorting to the Riemann theta functions.

Discrete coupled derivative nonlinear Schrödinger equations and their quasi-periodic solutions

A hierarchy of nonlinear differential-difference equations associated with a discrete isospectral problem is proposed, in which a typical differential-difference equation is a discrete coupled derivative nonlinear Schrödinger equation. With the help of the nonlinearization of the Lax pairs, the hierarchy of nonlinear differential-difference equations is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebraic curve, the Abel–Jacobi coordinates are introduced to straighten out the corresponding flows, from which quasi-periodic solutions for these differential-difference equations are obtained resorting to the Riemann-theta functions. Moreover, a (2+1)-dimensional discrete coupled derivative nonlinear Schrödinger equation is proposed and its quasi-periodic solutions are derived.

Methods for justifying arithmetic hypotheses and computer algebra

Computer algebra methods for justifying hypotheses in arithmetic geometry are considered. The specific feature of these methods is that they are designed for parametric problems. The methods developed can be used for solving computational problems and justifying hypotheses on equidistribution, as well as for the theory of algebraic curves over finite fields.

Explicit Solutions of the 2+1-Dimensional Modified Toda Lattice through Straightening out of the Relativistic Toda Flows

The 2+1-dimensional modified Toda lattice is decomposed into solvable ordinary differential equations with the help of the 1+1-dimensional relativistic Toda lattices. Based on the decomposition and the theory of algebraic curve, the straightening out of various flows, including the continuous flow and discrete flow, is exactly given through the introduced Abel–Jacobi coordinates. The explicit theta function solutions for the 2+1-dimensional modified Toda lattice are obtained explicitly.

Decomposition and straightening out of the coupled mixed nonlinear Schrödinger flows

The nonlinearization approach of Lax pairs is extended to the investigation of a soliton hierarchy proposed by Wadati, Konno and Ichikawa, in which the first nontrivial equation is the coupled mixed nonlinear Schrödinger equation. Under a constraint between the potentials and eigenfunctions, solutions of the soliton hierarchy are decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the class of finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Based on the decomposition and the theory of algebraic curve, the Abel–Jacobi coordinates are introduced to straighten out the corresponding flows. As an application, the compatible solutions of the various flows in Abel–Jacobi coordinates are explicitly obtained.

Multipolar vortices and algebraic curves

This paper demonstrates that analytical solutions to the steady two-dimensional Euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains. The solutions consist of a finite set of line vortices superposed on finite-area patches of uniform vorticity. By way of example, new solutions are presented in which the support of the vorticity is a quintuply connected vortex patch with five vorticity maxima modelling a symmetric pentapolar vortex consisting of a central core vortex, four satellite vortices and four enclosed zones of irrotational fluid between the core and the satellite vortices. The method of construction is amenable to the derivation of exact solutions of the two-dimensional Euler equations having vorticity distributions of even greater geometrical complexity.

Squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell

The theory of algebraic curves and quadrature domains is used to construct exact solutions to the problem of the squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell. The solutions are exact in that they can be written down in terms of a finite set of time-evolving parameters. The method is very general and applies to fluid domains of any finite connectivity. By way of example, the evolution of fluid domains with two and four air holes are calculated explicitly. For simply connected domains, the squeeze flow problem is well posed. In contrast, the squeeze flow problem for a multiply connected domain is not necessarily well-posed and solutions can break down in finite time by the formation of cusps on the boundaries of the enclosed air holes.

States and curves of five-dimensional gauged supergravity

We consider the sector of N=8 five-dimensional gauged supergravity with nontrivial scalar fields in the coset space SL(6, R)/SO(6) , plus the metric. We find that the most general supersymmetric solution is parametrized by six real moduli and analyze its properties using the theory of algebraic curves. In the generic case, where no continuous subgroup of the original SO(6) symmetry remains unbroken, the algebraic curve of the corresponding solution is a Riemann surface of genus seven. When some cycles shrink to zero size the symmetry group is enhanced, whereas the genus of the Riemann surface is lowered accordingly. The uniformization of the curves is carried out explicitly and yields various supersymmetric configurations in terms of elliptic functions. We also analyze the ten-dimensional type-IIB supergravity origin of our solutions and show that they represent the gravitational field of a large number of D3-branes continuously distributed on hyper-surfaces embedded in the six-dimensional space transverse to the branes. The spectra of massless scalar and graviton excitations are also studied on these backgrounds by casting the associated differential equations into Schrödinger equations with nontrivial potentials. The potentials are found to be of Calogero type, rational or elliptic, depending on the background configuration that is used.

On the Construction of Complete Sets of Geometric Invariants for Algebraic Curves

We provide a solution to the important problem of constructing complete independent sets of Euclidean and affine invariants for algebraic curves. We first simplify algebraic curves through polynomial decompositions and then use some classical geometric results to construct functionally independent sets of invariants. The results presented here represent some new contributions to algebraic curve theory that can be used in many application areas, such model-based vision, object recognition, graphics, geometric modeling, and CAD.

BIFURCATIONS OF CRITICAL PERIODS: CUBIC VECTOR FIELDS IN KAPTEYN'S NORMAL FORM

In this paper, we study the local bifurcations of critical periods in the neighbourhood of a non-degenerate centre of a cubic system in Kapteyn's normal form. We find that this system has no isochronous centres. We show that at most three local critical periods bifurcate from a centre at the origin. Moreover there exist perturbations which lead to the maximum number of critical periods. We get some of the results by reducing the question to that of transversal intersection of some algebraic surfaces, using computational algebraic geometry techniques (Theory of Resultant, Sturm's algorithm, Gröbner Bases), Ideal theory of Noetherian rings, ‘Transversality theory of algebraic curves.’ We show that no critical period bifurcates from infinity.

Algebraic-geometry codes

The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.

Local Bifurcations of Critical Periods in the Reduced Kukles System

AbstractIn this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gröbner bases, theory of the resultant, Sturm’s algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

Numerical curves and their applications to algebraic curves

Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme $N={n_1,..., n_s;n}$, i.e., the

Analysis of the offset to a parabola

Wei Lü has recently constructed a proper rational parameterization for the two-sided offset to a parabola—indicating that this locus, an irreducible curve of degree n = 6, is of genus 0. The theory of algebraic curves then requires that the parabola offset have 1 2 (n−1)(n−2) = 10 double points. One affine node and six affine cusps are readily identified by geometric arguments; we show that the remaining three double points are accounted for by a non-ordinary double point at infinity, having double points in its first and second neighborhoods.

On Curve Veering and Flutter of Rotating Blades

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to “curve veering”), or attract each other (leading to “frequency coalescence”). Our results are reached by using standard arguments from algebraic geometry—the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

Polynomial complexity algorithms for computational problems in the theory of algebraic curves

It is proved that the classical algorithm for constructing Newton-Puiseux expansions for the roots of polynomials using the method of Newton polygons is of polynomial complexity in the notation length of the expansion coefficients. This result is used, in the case of a ground field of characteristic O, to construct polynomial-time algorithms for factoring polynomials over fields of formal power series, and for fundamental computational problems in the theory of algebraic curves, such as curve normalization.

Construction of codes on arbitrary plane curves

In 1981, Goppa established an epoch-making method for constructing a code utilizing the theory of algebraic curves [1]. This is called the algebraic curve code. This paper develops the theory on an arbitrary curve, which may contain singular points and discusses the method of constructing the code. The method employed in this paper is to eliminate the singular point by applying the transformation called blowing-up and to construct the code on the nonsingular model. The number of genes of curves can be derived simultaneously. It is made easy to derive the bases of the linear spaces L(G) and Ω(G-D), as in the case of the nonsingular curve. The algebraic curve C considered in this paper is the irreducible projected plane curve or the algebraic curve which is obtained by gluing irreducible affine plane curves. In this case, the nonsingular model of C can also be represented as a gluing of irreducible affine plane curves. Then the method is applied to the hyperelliptic plane curve on which the number of rational points achieves the Hasse-Weil bound or certain hypernonelliptic plane curves. It is shown that the precise structure of the code can be examined.

Algebraic complexities and algebraic curves over finite fields

Algebraic schemes of computation of bilinear forms over various rings of scalars are examined. The problem of minimal complexity of these schemes is considered for computation of polynomial multiplication and multiplication in commutative algebras, and finite extensions of fields. While for infinite fields minimal complexities are known (Winograd, Fiduccia, Strassen), for finite fields precise minimal complexities are not yet determined. We prove lower and upper bounds on minimal complexities. Both are linear in the number of inputs. These results are obtained using the relationship with linear coding theory and the theory of algebraic curves over finite fields.

Parametric cubics as algebraic curves

This paper provides first a general background in the algebraic curve theory of cubics. The specific topics covered are double points, inflection points, tangent lines, three standard curves and the slope parameterization. Then parametric cubics are discussed in this context. Specific topics covered are double points, inflection points, tangent lines, degenerate cubic parameterizations, the inverse problem, and finding the implicit equation. All of these topics are presented so as to be easily translatable into computer algorithms.