has an acnode at the origin of , because it is equivalent to
and is non-negative when ≥ 1 and when . Thus, over the real numbers the equation has no solutions for 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 and 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.
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