The limit point and the T--function

Abstract

Let $\mathcal{P}(t)\in \mathbb{K}(t{)}^n$ be a rational parametrization of an algebraic space curve $\mathcal{C}$. 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 $H_{P_i}(s),i=1,\dots ,n$ are the fibre functions, and its 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.