Acnode
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An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve.
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.
[edit] See also
[edit] References
- Porteous, Ian (1994). Geometric Differentation. Cambridge University Press. ISBN 0-521-39063-X.
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