Computing Numerical Singular Points of Plane Algebraic Curves (Communications in Mathematical Research, 2021)
Given an irreducible plane algebraic curve of degree $d ≥ 3$, we computeits numerical singular points, determine their multiplicities, and count the numberof distinct tangents at each to decide whether the singular points are ordinary. Thenumerical procedures rely on computing numerical solutions of polynomial systemsby homotopy continuation method and a reliable method that calculates multipleroots of the univariate polynomials accurately using standard machine precision. It iscompletely different from the traditional symbolic computation and provides singularpoints and their related properties of some plane algebraic curves that the symbolicsoftware Maple cannot work out. Without using multiprecision arithmetic, extensivenumerical experiments show that our numerical procedures are accurate, efficient androbust, even if the coefficients of plane algebraic curves are inexact.
Keywords: numerical singular point; multiplicity; ordinary; homotopy continuation.