# 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 are commonly found in the study of algebraic curves over fields which are not algebraically closed, defined as the zero set of a polynomial of two variables. For example the equation

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

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

${\displaystyle y^{2}=x^{2}(x-1)}$

and ${\displaystyle x^{2}(x-1)}$ is non-negative when ${\displaystyle x}$ ≥ 1 and when ${\displaystyle x=0}$. Thus, over the real numbers the equation has no solutions for ${\displaystyle x<1}$ 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 ${\displaystyle \partial f \over \partial x}$ and ${\displaystyle \partial f \over \partial y}$ 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.