Singularities of plane algebraic curves

We give an exposition of some of the basic results on singularities of plane algebraic curves, in terms of polynomials and formal power series.     

Local intersections of plane algebraic curves

We determine the maximum punctual order of contact between two plane algebraic curves, of which one is reduced. We prove that, generically on the reduced curve, this quantity is always strictly bounded by the product of the degrees of the curves.

     

Real algebraic curves of constant width

Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases.     

Real intersection points of piecewise algebraic curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this work, we present an algorithm for computing the real intersection points of piecewise algebraic curves. It is primarily based on the interval zeros of the univariate interval polynomial in Bernstein form. An illustrative example is provided to show that the proposed algorithm is flexible.     

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 \({k \in \mathbb{N}}\) .     

Complex algebraic plane curves via their links at infinity

Summary Considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew, surprisingly little is known about the topology of affine algebraic plane curves. We topologically classify regular algebraic plane curves in complex affine 2-space using splice diagrams: certain decorated trees that code Puiseux data at infinity. (The regularity condition — that the curve be a typical fiber of its defining polynomial — can conjecturally be avoided.) We also show that the splice diagram determines such algebraic information as the minimal degree of the curve, even in the irregular case. Among other things, this enables algebraic classification of regular algebraic plane curves with given topology.     

On multiplicity of intersection point of two plane algebraic curves

The paper studies the multiplicity of intersecting point of two plane algebraic curves. The multiplicity is characterized by means of operators with partial derivatives. It is proved that if A is a point of multiplicity m for one of the curves and, a point of multiplicity n for the other curve, then the arithmetical multiplicity of the intersection (or the number of intersections) of the curves in A, is not less than mn and is equal to mn when the curves do not have common tangents at the point A.     

The minimal degree of plane models of algebraic curves and double coverings

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g ≠ 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.     

Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from operator theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but some of our results have purely algebraic formulation. We make use of Xia's theory of subnormal operators and of the previous results by the author in this direction. We also correct (in Section 5) some inaccuracies in the works of [D.V. Yakubovich, Subnormal operators of finite type I. Xia's model and real algebraic curves in C², Rev. Mat. Iberoamericana 14 (1998) 95-115; D.V. Yakubovich, Subnormal operators of finite type II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998) 623-681] by the author.