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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
:$f(x,y)=y^2+x^2+x^3=0;$
has an acnode at the origin of $mathbb\{R\}^2$, because it is equivalent to
:$y^2\; =\; -(x^2\; +\; x^3)$
and $x^2\; +\; x^3$ 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 $partial\; fover\; partial\; x$ and $partial\; fover\; partial\; y$ vanish. Further the Hessian matrix of second derivatives will be positive definite. Hence the function has a local minimum or local maximum.

Contents

See also

External links

References

See also

★ Singular point of a curve

★ Crunode

★ Cusp

★ Tacnode

External links

★ Diagram of an acnode [[1]]

References

★ Geometric Differentation, , Ian, Porteous, Cambridge University Press, 1994, ISBN 0-521-39063-X
1. http://eom.springer.de/I/i052770.htm
2. http://eom.springer.de/A/a130100.htm