Acnode



An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point  and hermit point.

For example the equation
 * $$f(x,y)=y^2+x^2-x^3=0$$

has an acnode at the origin, because it is equivalent to
 * $$y^2 = x^2 (x-1)$$

and $$x^2(x-1)$$ is non-negative only when $$x$$ ≥ 1 or $$x = 0$$. Thus, over the real numbers the equation has no solutions for $$x < 1$$ except for (0, 0).

In contrast, over the complex numbers the origin is not isolated since square roots of negative real numbers exist. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point.

An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives $$\partial f\over \partial x$$ and $$\partial f\over \partial y$$ vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite, since the function must have a local minimum or a local maximum at the singularity.