The limit point and the T-function. (September 2019) - Blasco, Angel; Pérez-Díaz, Sonia
Abstract: Let P ( t ) ∈ P 2 ( K ( t ) ) be a rational projective parametrization of a plane curve C . In this paper, we introduce the notion of limit point, P L, of P ( t ), and some remarkable properties of P L are obtained. In particular, if the singularities of C are P 1, …, P n and P L (all of them ordinary) and their respective multiplicities are m 1, …, m n and m L, we show that T ( s ) = ∏ i = 1 n H P i ( s ) m i − 1 H P L ( s ) m L − 1, where T ( s ) is the univariate resultant of two polynomials obtained from P ( t ), and H P 1 ( s ), …, H P n ( s ), H P L ( s ) are the fibre functions of the singularities. The fibre function of a point P is a polynomial H P ( 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.