%0 Journal Article
%A Pérez Díaz, Sonia
%A Blasco Lorenzo, Ángel
%T An in depth analysis, via resultants, of the singularities of a parametric curve
%D 2019
%@ 0167-8396
%U http://hdl.handle.net/10017/41542
%X Let C be an algebraic space curve defined by a rational parametrization P(t)∈K(t)ℓ, ℓ≥2. In this paper, we consider the T-function, T(s), which is a polynomial constructed from P(t) by means of a univariate resultant, and we show that T(s) contains essential information concerning the singularities of C. More precisely, we prove that T(s)=∏i=1nHPi(s), where Pi, i=1,…,n, are the (ordinary and non-ordinary) singularities of C and HPi, i=1,…,n, are polynomials, each of them associated to a singularity, whose factors are the fiber functions of those singularities as well as those other belonging to their corresponding neighborhoods. That is, HQ(s)=HQ(s)m−1∏j=1kHQj(s)mj−1, where Q is an m-fold point, Qj,j=1,…,k, are the neighboring singularities of Q, and mj,j=1,…,k, are their corresponding multiplicities (HP denotes the fiber function of P). Thus, by just analyzing the factorization of T, we can obtain all the singularities (ordinary and non-ordinary) as well as interesting data relative to each of them, like its multiplicity, character, fiber or number of associated tangents. Furthermore, in the case of non-ordinary singularities, we can easily get the corresponding number of local branches and delta invariant.
%K Rational parametrization
%K Singularities of an algebraic curve
%K Multiplicity of a point
%K Ordinary and non-ordinary singularities
%K T-function
%K Fiber function
%K Matemáticas
%K Mathematics
%~ Biblioteca Universidad de Alcala