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.^[1]

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 $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ 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.