Building Abelian Functions with Generalised Baker-Hirota Operators

Decoding by rank-2 bundles over plane quartics

Riemann continued to develop his own ideas, extending them to multiply-connected domains defined by algebraic curves. He presented them in public when he lectured on complex functions, in particular elliptic and Abelian functions , in 1855/56 and again in 1861/62, and he published them in his remarkable paper on abelian functions in 1857.KeywordsComplex Function TheorypolesPoleMany-valued NaturePeriod ParallelogramPuiseuxThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition. We say that a subscheme Z o f P 3 has maximal rank iffol every integer t the restriction map ~z( t ) :H(P 3, Op(t))~H~ 0z(t)) is either injective or surjeetive. It is useful to know that a curve C in pa has maximal rank since in this case for every k we know completely the dimension of the vector space of surfaces of degree k containing C. For example if C is a canonical curve of genus g with maximal rank, it

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d-1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

In this chapter we present the basic facts about algebraic curves (i.e. projective varieties of dimension 1) which will be needed for our study of elliptic curves. (Actually, since elliptic curves are curves of genus 1, one of our tasks will be to define the genus of a curve.) As in Chapter I, we give references for those proofs which are not included. There are many books where the reader can find more material on the subject of algebraic curves, for example [Har, Ch. IV], [Sha 2], [G–H, Ch. 2], [Wa].KeywordsSmooth CurveElliptic CurfFunction FieldAlgebraic CurfSmooth CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.

Introduction. Let k be an algebraically closed ground field of characteristic zero. The following result is well-known (see Severi [1], Anhang F, p. 317). If En d is a complete (i.e., maximal) irreducible algebraic system, defined over k, of plane algebraic (not necessarily irreducible) curves of a given order n, such that the general curve C* of En d /k has d nodes (and no other singularities), then the dimension of En d is equal to 3n + p 1, where p = ((n 1)(n 2)/2) d is the genus of C*. We note that 3n + p 1 is equal to (n (n + 3)/2) d. Here N = (n(n + 3)/2) is the dimension of the (linear) system of all plane curves of order n. This is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes (in non-assigned position) imposes d independent algebraic (non-linear) conditions on the curve. The above result is easily proved and will be included by us in a more general theorem, the proof of which is the main object of this paper (Theorem 2, Section 3) and which deals with irreducible algebraic systems E of curves of order n in which the general curve C* of E/k has arbitrary singularities (but is free from multiple components; equivalently: C* is a reduced curve). This theorem asserts that if p is the genus of C* (the definition of the effective genus of a reducible-but reduced-curve is given in Section 3, formula (2)), then dim E c 3n + p 1, with equality if and only if C* has only nodes as singularities. It is easily seen (see Severi, loc. cit.) that if 0 c d ' (n(n 1)/2) then there always exists curves of order n having d nodes (the maximum

It is somnewhat surprising that the systematic evaluation of the heights of rational points on a curve and on its jacobian variety and particularly of their relation to each other should yield any new information. Nonetheless this appears to be the case and the result is described in this article. Although the main theorem is not even a special case of the very fascinating conjecture of Mordell, still it is an estimate that already reveals that rational points on curves of genus at least 2 are much harder to come by than on curves of genus 0 or 1. It is a quantitative limitation on the heights of such points which is well-known to be false in the case of genus 0 or 1. Incidentally, there is a good explanation why an estimate of this type can be obtained so cheaply, whereas Mordell's conjecture itself could not: namely, results obtained by our methods will more or less automatically apply to the analogous funetion case [where the ground field is a function field in one variable over a finite field, rather than an algebraic number field]. And in this case, unless further restrictions are imposed, there are curves of any genus with an infinite number of rational poilnts whose heights increase exactly at the rate which we will find. Let k be an algebraic number field of finite degree over Q. Let C be a non-singular projective curve over k of genus g at least 2. Mordell's conjecture asserts that the set of k-rational points on C is finite. Now suppose

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

Cyclic étale coverings of generic curves and ordinariness of dormant opers

The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation (Gaudry); for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors (Diem, Kochinke). Recently, Sarkar and Singh have proposed a sieving technique, inspired by an earlier work of Joux and Vitse, to speed up the relation search in the hyperelliptic case. We give a new description of this technique, and show that this new formulation applies naturally to the non-hyperelliptic case with or without large prime variations. In particular, we obtain a speed-up by a factor approximately 3 for the relation search in Diem and Kochinke’s methods.

In the kinematic analysis of the coupler curve, a parametric equation with nine parameters, which are dependent on the mechanism dimensions, is commonly used. In this study, we extend the analysis of a four-bar mechanism by introducing a 10th parameter, referred to as the closure parameter. Additionally, we present a comprehensive, practical, and fast simulation that displays real-time values of 16 kinematic variables throughout motion. The accuracy of this simulation is validated through comparisons with a manually constructed mechanism and tested successfully across various mechanism configurations. Another novelty of this work is the introduction of the general coupler curve equation curve fitting method (GCCE-CFM), which enables fast and practical kinematic synthesis of the coupler curve. This method derives explicit expressions for the 16 coefficients of the sixth-degree general coupler curve equation (GCCE), also known as the algebraic coupler curve equation, in terms of the mechanism parameters. Subsequently, real-time curve fitting is applied to 40 points sampled from a known coupler curve, successfully recovering the mechanism dimensions with high accuracy in just a few seconds. The proposed model achieves a high-precision curve fit with a root-mean-square error (RMSE) of 6.85e–5, demonstrating its effectiveness in accurately capturing the target coupler path.

This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bezier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.

In the previous chapter, a detailed description of the algorithms for the ‘algcurves’ package in Maple was presented. As discussed there, the package is able to handle general algebraic curves with coefficients given as exact arithmetic expressions, a restriction due to the use of exact integer arithmetic. Coefficients in terms of floating point numbers, i.e., the representation of decimal numbers of finite length on a computer, can in principle be handled, but the floating point numbers have to be converted to rational numbers. This can lead to technical difficulties in practice. One also faces limitations if one wants to study families of Riemann surfaces, where the coefficients in the algebraic equation defining the curve are floating point numbers depending on a set of parameters, i.e., if one wants to explore modular properties of Riemann surfaces as in the examples discussed below. An additional problem in this context can be computing time since the computation of the Riemann matrix uses the somewhat slow Maple integration routing. Thus, a more efficient computation of the Riemann matrix is interesting if one wants to study families of Riemann surfaces or higher genus examples which are computationally expensive.KeywordsRiemann SurfaceBranch PointTheta FunctionAlgebraic CurveAlgebraic CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.