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 f i ( x 1 , x 2 , x 3 ) 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 C p . In this work, we leverage this established relationship to delineate the asymptotic behavior of C by examining the asymptotes of C p . Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of C , given that the asymptotes of C p have been ascertained. • Let C represent an irreducible algebraic space curve defined by the real polynomials f i ( x 1 , x 2 , x 3 ) , i = 1 , 2 . • A birational relationship always exists between the points on C and those on an associated irreducible plane curve C p . • We use this relationship to analyze the asymptotic behavior of C by examining the asymptotes of C p . • We introduce a novel algorithm designed to efficiently compute the asymptotes of C by studying the asymptotes of C p .