Acnode

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

For example the equation

has an acnode at the origin, because it is equivalent to

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 ∂ f ∂ x and ∂ f ∂ 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.