The limit point and the T-function

Abstract

Let P(t) ϵ P2 (K(t)) be a rational projective parametrization of a plane curve C. In this paper, we introduce the notion of limit point, PL, of P(t), and some remarkable properties of PL are obtained. In particular, if the singularities of C are P1, . . . , Pn and PL (all of them ordinary) and their respective multiplicities are m1, . . . , mn and mL, we show that T(s) = n i=1 HPi (s) m-1HPL (s) mL-1 , where T(s) is the univariate resultant of two polynomials obtained from P(t), and HP1 (s), . . . , HPn (s), HPL (s) are the fibre functions of the singularities. The fibre function of a point P is a polynomial HP (s) whose roots are the fibre of P. Thus, a complete classification of the singularities of a given plane curve, via the factorization of a resultant, is obtained.