# Acnode

An acnode at the origin (curve described in text)

An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term.[1]

Acnodes commonly occur when studying algebraic curves over fields which are not algebraically closed, defined as the zero set of a polynomial of two variables. For example the equation

$f(x,y)=y^2+x^2-x^3=0\;$

has an acnode at the origin of $\mathbb{R}^2$, because it is equivalent to

$y^2 = x^2 (x-1)$

and $x^2(x-1)$ is non-negative when $x$ ≥ 1 and when $x = 0$. 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.

An acnode is a singularity of the function, where 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. Hence the function has a local minimum or a local maximum.