RT info:eu-repo/semantics/article
T1 An in depth analysis, via resultants, of the singularities of a parametric curve
A1 Pérez Díaz, Sonia
A1 Blasco Lorenzo, Ángel
K1 Rational parametrization
K1 Singularities of an algebraic curve
K1 Multiplicity of a point
K1 Ordinary and non-ordinary singularities
K1 T-function
K1 Fiber function
K1 Matemáticas
K1 Mathematics
AB 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.
PB Elsevier
SN 0167-8396
YR 2019
FD 2019-01-01
LK http://hdl.handle.net/10017/41542
UL http://hdl.handle.net/10017/41542
LA eng
NO Agencia Estatal de Investigación
DS MINDS@UW
RD 09-dic-2023