Multiple view geometry of non-planar algebraic curves

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

The Curvatures of the Polar Curves of a General Algebraic Curve

In Hough [1], Duda and Hart [2], and Griffith [3] procedures were proposed for detecting lines in pictures and in [2] Duda and Hart extended their method for more general algebraic curve, fitting. This correspondence shows how this method can be used to detect any given curve in a specific orientation. The procedure presented here con be easily implemented and can be efficiently implemented in a parallel machine.

The Jacobian of a very general complex algebraic curve of genus at least 3 contains an algebraic cycle called the Ceresa cycle that is homologically trivial but algebraically nontrivial. Zharkov defined in analogy the tropical Ceresa cycle of a metric graph and proved a similar result for very general tropical curves overlying the complete graph on four vertices. We extend this result by considering a related, ‘universal’ invariant of the underlying graph called the Ceresa period; we show that having trivial Ceresa period has a forbidden minor characterization that coincides with the graph being of hyperelliptic type.

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.

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.

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly limit the types of involved exact operations, that is, our algorithms only use resultant and gcd computations as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the use of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system.We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

The Kruppa equations contain all the available geometric information and enable to recover the epipolar geometry from uncalibrated perspective views. In this paper, the matrix system FCF/sup T/=/spl lambda/e/sub 1//sup x/Ce/sub 1//sup xT/ is studied (the Kruppa equations are its deformation), and the general solution form of C is presented. In the end, some analyses and conclusions of the Kruppa equations are proposed.

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

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

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.

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 Uniformization Theorem due to Koebe and Poincare implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

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