A tropical curve is a graph with specified edge lengths, some of which may be infinite. Various facts and attributes about algebraic curves have analogs for tropical curves. In this article, we focus on divisors and linear series, and prove the Riemann-Roch formula for divisors on tropical curves. We describe two ways in which algebraic curves may be transformed into tropical curves: by aboemas...

We present an approach that extends the BentleyOttmann sweep-line algorithm [3] to the exact computation of the topology of arrangements induced by non-singular algebraic curves of arbitrary degrees. Algebraic curves of degree greater than 1 are difficult to handle in case one is interested in exact and efficient solutions. In general, the coordinates of intersection points of two curves are no...

Introduction The existence of singular algebraic curves with given invariants and given set of singular-ities and, at the same time, the study of (equisingular) families of such curves is a very old, but still attractive and widely open, problem. Already, at the beginning of the 20th century, the foundations were made in the works of Pl ucker, Severi, Segre and Zariski. In the sequel the theory...

January 23, 2013 Frances Kirwan’s book Complex algebraic curves is an excellent introduction to complex algebraic curves (see [5]). Whenever possible I have included a page reference to the book, in the form [5]. Another beautiful book on this subject is Rick Miranda’s book Algebraic curves and Riemann surfaces (see [6]). To get a feeling what is higher-dimensional complex algebraic geometry, s...

Algebraic curves and surfaces have been studied intensively in algebraic geometry for centuries. Thus, there exists a huge amount of theoretical knowledge about these geometric objects. Recently, algebraic curves and surfaces play an important and ever increasing rôle in computer aided geometric design, computer vision, computer aided manufacturing, coding theory, and cryptography, just to name...

An algebraic curve is defined as the zero set of a polynomial in two variables. Algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics. The main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data. Usually, arguments against using algebraic curves involve references to mathemati...

In this paper we give an overview of the symbolic computation of rational parametrizations for algebraic curves. Parametrization of algebraic curves has been a topic in algebraic geometry for a long time and there are basically two approaches to the problem. The first approach is a geometric one, whereby a parametrization is determined from the intersection points of the given curve with a cert...