Resultants and Singularities of Parametric Curves

Abstract

Let $\mathcal{C}$ be an algebraic space curve defined parametrically by $\mathcal{P}(t)\in \mathbb{K}(t{)}^n,n\ge 2$. In this paper, we introduce a polynomial, the T--function, $T(s)$, which is defined by means of a univariate resultant constructed from $\mathcal{P}(t)$. We show that $T(s)={\prod }_{i=1}^n{H}_{P_i}(s{)}^{m_i-1}$, where $H_{P_i}(s),i=1,\dots ,n$ are polynomials (called the fibre functions) whose roots are the fibre of the ordinary singularities $P_i\in \mathcal{C}$ of multiplicity $m_i,i=1,\dots ,n$. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.

References in corpus (1)

• Asymptotes of Space Curves