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

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

y² = − (x² + x³)

and x² + x³ 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 and vanish. Further the Hessian matrix of second derivatives will be positive definite. Hence the function has a local minimum or local maximum.

See also

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

References

• Porteous, Ian (1994). Geometric Differentation. Cambridge University Press. ISBN 0-521-39063-X. 

Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Acnode". Image Credit.

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