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On a singularity appearing in the multiplication of polynomials. (English) Zbl 1461.14007

The space of monic polynomials of degree \(n\) with coefficients in a real or complex field \(\mathbf{K}\) is denoted by \(\mbox{MP}(\mathbf{K},n)\), which can be identified with \(\mathbf{K}^n\), since a polynomial in \(\mbox{MP}(\mathbf{K},n)\) is given by \(n\) coefficients. The authors are interested in understanding the properties of the map: Mult: \(\mbox{MP}(\mathbf{K},n)\times \mbox{MP}(\mathbf{K},m)\to \mbox{MP}(\mathbf{K},n+m)\), which multiplies two polynomials and it can be identified as a mapping from \(\mathbf{K}^{n+m}\) to itself.
The authors study the singularity type of Mult when the two polynomials have a common double root. They start to consider two polynomials of degree 2 with one single root which is common and double in both of them. After they consider two polynomials with only one root which is common, double in one of them and of multiplicity \(k \geq 2\) in the other one. Finally the authors combine all the cases known to give a statement about pairs of polynomials whose greater common divisor has only simple or double roots.

MSC:

14B05 Singularities in algebraic geometry
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
58K05 Critical points of functions and mappings on manifolds

Citations:

Zbl 1200.32018
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