Infinity Branches and Asymptotic Analysis of Algebraic Space Curves: New Techniques and Applications

Let C represent an irreducible algebraic space curve defined by the real polynomials fi(x1, x2, x3) for i = 1, 2. It is a recognized fact that a birational relationship invariably exists between the points on C and those on an associated irreducible plane curve, denoted as Cp. In this work, we leverage this established relationship to delineate the asymptotic behavior of C by examining the asymptotes of Cp. Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of C, given that the asymptotes of Cp have been ascertained.