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A regularization approach for estimating the type of a plane curve singularity. (English) Zbl 1298.14066
The paper addresses the problem of analyzing the local topology of each singularity of a plane complex algebraic curve defined by a squarefree polynomial. The local topology is computed by means of a symbolic-numeric algorithm that uses tools from knot theory, namely the Alexander polynomial and the delta-invariant of the singularity. As a novelty, the paper deals not only with the exact case, but also with the case when the coefficients of the polynomial defining the curve are known up to a finite tolerance. In this case, the computation of the local topology is an ill-posed problem, in the sense that tiny changes in the input may result in qualitative changes in the output. In the inexact case, the paper also provides a parameter choice rule that satisfies the convergence for noisy data property. This implies that as the noise in the data reduces to zero, the output provided by the algorithm converges to the exact answer. Furthermore, the algorithm is implemented in a new software package called CENOM3CK, written in the Axel free algebraic geometric modeler and in the Mathemagix free computer algebra system. A potential application of the algorithm is the computation of the genus of an implicit algebraic curve, whose coefficients are known up to a fixed tolerance.

14Q05 Computational aspects of algebraic curves
14B05 Singularities in algebraic geometry
68W30 Symbolic computation and algebraic computation
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
Full Text: DOI
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