Determining the asymptotic family of an implicit curve
This paper considers the problem of determining the asymptotic family of a plane algebraic curve \(\mathcal C\) which is assumed to have only regular points at infinity. The asymptotic family of \(\mathcal C\) is the set of algebraic plane curves that have the same asymptotic behaviour as \(\mathcal C\). The main result states that these curves form the vector space of all algebraic plane curves of the degree \(d = \deg \mathcal C\) whose defining equations share the homogeneous forms of degree \(d\) and of degree \(d-1\). This generalizes results of [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, Comput. Aided Geom. Des. 31, No. 2, 81--96 (2014; Zbl 1301.14033)] for curves with only one regular infinity branch. The characterization of asymptotic families simplifies the computation of generalized asymptotes of \(\mathcal C\) (plane algebraic curves of minimal degree, one for each infinity branch, whose distance to \(\mathcal C\) converges to zero as the curves tend to infinity) as it is possible to ignore the irrelevant coefficients of the low-degree homogeneous forms. It also allows to compute in a straightforward way an asymptotic family from its set of regular \(g\)-asymptotes by simply multiplying the \(g\)-asymptotes' equations and extracting the two homogeneous forms of highest degree. This paper is self-contained with precise definitions of central concepts, collections of previous results, and examples illustrating results and theories. Proofs are collected in a concluding section. An obvious question for future research is the generalization of the results to the case of curves with non-regular points at infinity.
scientific article; zbMATH DE number 7596221
scientific article