On a singularity appearing in the multiplication of polynomials

Journal of Singularities
volume 22 (2020), 205-214

Received: 28 February 2019. Received in revised form: 20 August 2020.

DOI: 10.5427/jsing.2020.22n

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Abstract:

The multiplication of monic polynomials of degrees n and m defines a mapping R^{n+m} -> R^{n+m}. Singularities of this mapping at a point corresponding to two polynomials (P,Q) appear when the two polynomials have a common root. In Ch-LdM, it was shown that, when every such common root is simple in one of the polynomials, the singularity type can be described using swallowtail singularities whose geometry is well understood. In this paper we consider the case where there are common double roots. We start with the minimal possible situation where both polynomials are of degree 2, and give a normal form for the singularity that allows us to describe its geometry quite thoroughly. This normal form is then extended to other polynomial pairs with only one common multiple root which is a double root in one of them. Finally we give a general statement for pairs whose greater common divisor has only single or double roots.

Author(s) information:

Santiago López de Medrano
Instituto de Matemáticas
UNAM
email: santiago@matem.unam.mx

Enrique Vega Castillo
Facultad de Ciencias
UNAM
email: enriquevegac@ciencias.unam.mx