The limit point and the T--function

Abstract

Let ${\cal P}(t)\in {\Bbb K}(t)^{n}$ be a rational parametrization of an algebraic space curve $\cal C$. In this paper, we introduce the notion of limit point, $P_L$, of the given parametrization $\mathcal{P}(t)$, and some remarkable properties of $P_L$ are obtained. In addition, we generalize the results in \cite{MyB-2017} concerning the T--function, $T(s)$, which is defined by means of a univariate resultant. More precisely, independently on whether the limit point is regular or not, we show that $T(s)=\prod_{i=1}^n H_{P_i}(s)^{m_i-1}$. The polynomials $H_{P_i}(s),\,i=1,\ldots,n$ are the fibre functions, and its roots are the fibre of the ordinary singularities $P_i\in {\cal C}$ of multiplicity $m_i,\,i=1,\ldots,n$. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.