Algebraic Curves in The Plane
Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form f(x, y)=0, where f is a polynomial function f:R2→R. If f is expanded as
If the origin (0, 0) is on the curve then a0=0. If b1≠0 then the implicit function theorem guarantees there is a smooth function h so that the curve has the form y=h(x) near the origin. Similarly, if b0≠0 then there is a smooth function k so that the curve has the form x=k(y) near the origin. In either case, there is a smooth map from R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin
so the curve is non-singular or regular at the origin if at least one of the partial derivatives of f is non-zero. The singular points are those points on the curve where both partial derivatives vanish,
Read more about this topic: Singular Point Of A Curve
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