Acnode

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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] ^[2]

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 3)

and $x 2 + x 3$ is positive for $x > - 1$ , except 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 local maximum.