On the square-freeness of the offset equation to a rational planar curve

Díaz Toca, Gema M.; Alcázar, Juan Gerardo; Caravantes, Jorge

Publication:
On the square-freeness of the offset equation to a rational planar curve

dc.contributor.author	Díaz Toca, Gema M.
dc.contributor.author	Alcázar, Juan Gerardo
dc.contributor.author	Caravantes, Jorge
dc.contributor.department	Ingeniería y Tecnología de Computadores
dc.date.accessioned	2024-01-30T17:43:26Z
dc.date.available	2024-01-30T17:43:26Z
dc.date.issued	2018-04-04
dc.description	©2018. This manuscript version is made available under the CC-BY 4.0 license http://creativecommons.org/licenses/by/4.0/ This document is the Accepted, version of a Published Work that appeared in final form in International Journal of Algebra and Computation (iJAC). To access the final edited and published work see https://doi.org/10.1142/S0218196718500194	es
dc.description.abstract	It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the non-extraneous component of this resultant has at most two irreducible factors. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be com- puted in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the off- set has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the non-extraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.	es
dc.format	application/pdf	es
dc.format.extent	18	es
dc.identifier.citation	International Journal of Algebra and ComputationVol. 28, No. 03, pp. 395-409 (2018)
dc.identifier.doi	https://doi.org/10.1142/S0218196718500194
dc.identifier.issn	1793-6500
dc.identifier.issn	0218-1967
dc.identifier.uri	http://hdl.handle.net/10201/138177
dc.language	eng	es
dc.publisher	World Scientific	es
dc.relation	The authors were supported by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2014-54141-P.	es
dc.rights	info:eu-repo/semantics/openAccess	es
dc.rights	Atribución 4.0 Internacional	*
dc.rights.uri	http://creativecommons.org/licenses/by/4.0/	*
dc.subject	Offset curves	es
dc.subject	Planar rational curves	es
dc.subject	Squarefree factorization	es
dc.subject	Implicit equations	es
dc.title	On the square-freeness of the offset equation to a rational planar curve	es
dc.type	info:eu-repo/semantics/article	es
dspace.entity.type	Publication	es