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Volume 28, Issue 2
Computing Numerical Singular Points of Plane Algebraic Curves

Zhongxuan Luo, Er-Bao Feng & Wenyu Hu

Commun. Math. Res., 28 (2012), pp. 146-158.

Published online: 2021-05

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  • Abstract

Given an irreducible plane algebraic curve of degree $d ≥ 3$, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.

  • Keywords

numerical singular point, multiplicity, ordinary, homotopy continuation.

  • AMS Subject Headings

65D99, 13D15, 14Q05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-28-146, author = {Zhongxuan and Luo and and 18483 and and Zhongxuan Luo and Er-Bao and Feng and and 18485 and and Er-Bao Feng and Wenyu and Hu and and 18486 and and Wenyu Hu}, title = {Computing Numerical Singular Points of Plane Algebraic Curves}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {28}, number = {2}, pages = {146--158}, abstract = {

Given an irreducible plane algebraic curve of degree $d ≥ 3$, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19057.html} }
TY - JOUR T1 - Computing Numerical Singular Points of Plane Algebraic Curves AU - Luo , Zhongxuan AU - Feng , Er-Bao AU - Hu , Wenyu JO - Communications in Mathematical Research VL - 2 SP - 146 EP - 158 PY - 2021 DA - 2021/05 SN - 28 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19057.html KW - numerical singular point, multiplicity, ordinary, homotopy continuation. AB -

Given an irreducible plane algebraic curve of degree $d ≥ 3$, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.

Zhongxuan Luo, Er-Bao Feng & Wenyu Hu. (2021). Computing Numerical Singular Points of Plane Algebraic Curves. Communications in Mathematical Research . 28 (2). 146-158. doi:
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