Singular points of algebraic curves

Given an irreducible algebraic curve f(x,y)=0 of degree n≥3 with rational coefficients,we describe algorithms for determinig whether the curve is singular, and if so, isolating its singular points, computing their multiplicities, and counting the number of distinct tangents at each. The algorithms require only rational arithmetic operations on the coefficients of f(x,y)=0, and avoid the need for more abstract symbolic representations of the singular point coordinates. © 1990, Academic Press Limited. All rights reserved.