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Acnode

## Summary

An acnode at the origin (curve described in text)

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".[1]

For example the equation

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

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

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

and ${\displaystyle x^{2}(x-1)}$ is non-negative only when ${\displaystyle x}$ ≥ 1 or ${\displaystyle x=0}$. 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. 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 ${\displaystyle \partial f \over \partial x}$ and ${\displaystyle \partial f \over \partial y}$ vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite, since the function must have a local minimum or a local maximum at the singularity.