Topology of real algebraic curves

The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.     

Clifford's inequality for real algebraic curves

We improve Clifford's Inequality for real algebraic curves. As an application we improve Harnack's Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998.     

Sums of squares on real algebraic curves

Given an affine algebraic variety V over with real points V() compact and a non-negative polynomial function f[V] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in [V]. We then specialize to the case where V is a curve. The notion of virtual compactness is introduced, and it is shown that in the local-global principle, compactness of V() can be relaxed to virtual compactness. The irreducible curves on which every non-negative polynomial is a sum of squares are classified. All results are extended to the more general framework of preorders. Moreover, applications to the K-moment problem from analysis are given. In particular, Schmüdgens solution of the K-moment problem for compact K is extended, for dim (K)=1, to the case when K is virtually compact. Mathematics Subject Classification (1991):14P05, 11E25, 14H99, 14P10, 44A60. Dedicated to Eberhard Becker on the occasion of his 60th birthday     

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve ??; namely, via analytic continuation of unit speed parameterization of the real locus ${\Gamma_\mathbb{R}}$ . Globally, the metric ${\rho=|Q|=|q(z)|dzd\bar{z}}$ is given by the meromorphic quadratic differential Q on ?? induced by the standard complex form dx ²?+?dy ² on ${\mathbb{C}^2=\{(x,y)\}}$ . By considering basic properties of Q, we show that the condition for local arc length parameterization along ${\Gamma_\mathbb{R}}$ to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.     

Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X^′ between real algebraic curves where X is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X^′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X^′.     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

Self-duality equation: Monodromy matrices and algebraic curves

It is shown that the general local solution of the self-duality equation with SU(1,1) and SU(2) gauge groups is associated with some algebraic curve with moving branch points if the related “monodromy matrix” is rational. The “multisoliton” solutions including monopoles and instantons, correspond to degenerate curves when the branch cuts collapse to double points. Bibliography:18 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 197–216. Translated by D. A. Korotkin.     

On the realization problem of real algebraic plane curves as Hessian curves

In this paper we study some problems in Hessian topology. We prove that certain real plane curves satisfy the requirements of the Hessian curve of a differential function. The real plane curves we consider are those with k outer ovals and also those which only have one nest of depth k, with \({k \in \mathbb{N}}\) .     

Computation of roots of real and complex matrices

An algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix. All required computations are in the real domain. The method is based on the Newton-Raphson algorithm and is capable of finding roots even when the matrix is defective. Computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available.     

Vector bundles on a product of real algebraic curves

We study complex vector bundles on a product of nonsingular real algebraic curves.