There are several methods for solving a nonlinear algebraic equation having roots of a given multiplicity m. Here we compare a family of Laguerre methods of order three as well as two others of the same order and show that Euler–Cauchy’s method is best. We discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction. Published by Elsevier Inc.

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces:m-hypermaps andm-constellations. Form = 2 they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs combine a bijective approach, generating series techniques re...

In this paper, we study the moment map image of algebraic curves in toric surfaces. We are particularly interested in the situations that we are able to perturb the moment map so that the moment map image of the algebraic curve is a graph. To put our problem into proper context, let’s start with CP. Consider the natural real n-torus (T ) action on CP given by e(x) = (ex1, e x2, · · · , exn). Pa...

We give the best possible upper bound on the number of exceptional values and totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic Bryant surfaces and some partial results on the Osserman problem for algebraic Bryant surfaces. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

The map x y p x y x of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability The plane is a union of smooth disjoint invariant curves of the map However the integral has not explicitly been known In the current paper we will show that the map does not have an algebraic integral i e there is no non constant function F x y such that F ...

In this article, we study local holomorphic conformal embeddings from B into B1 × · · · ×BNm with respect to the normalized Bergman metrics. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic...

This paper is a continuation of [4, 9, 13]. To classify algebraic varieties is one of the goals in algebraic geometry. One way to study a given variety is to understand the behavior of its pluricanonical maps. The objects concerned here are complex projective 3-folds of general type over C. Let X be such an object and denote by φm the m-th pluricanonical map of X , which is the rational map ass...

In this article, we study local holomorphic isometric embeddings from Bn into BN1 × · · · × BNm with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and M...

Elimination is a basic algebraic operation which geometrically corresponds to projections. This article describes using the numerical algebraic geometric concept of witness sets to compute the projection of an algebraic set. The ideas described in this article apply to computing the image of an algebraic set under any linear map.