The T-function of a parametric curve

Abstract

In this paper, we introduce the T–function, T(s), which is a polynomial defined by means of a univariate resultant constructed from a given parametrization P(t) ∈ K(t) n , n ≥ 2 of an algebraic space curve C. It is shown that T(s) = Qn i=1 HPi (s) mi−1 , where HPi (s), i = 1, . . . , n are polynomials (the fibre functions) whose roots are the fibre of the ordinary singularities Pi ∈ C of multiplicity mi , i = 1, . . . , n of C. Therefore, a complete classification of the singularities of C, via the factorization of a resultant, is obtained.