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.

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

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

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.

See also

• Singular point of a curve
• Crunode
• Cusp
• Tacnode

References

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