Acnode

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. [ [http://eom.springer.de/I/i052770.htm Encyclopaedia of Mathematics, Springer Online Reference Works.] ] [ [http://eom.springer.de/A/a130100.htm Encyclopaedia of Mathematics, Springer Online Reference Works] ]

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.

ee also

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

References

*cite book |last=Porteous |first=Ian |title=Geometric Differentation |year=1994 |publisher=Cambridge University Press |id=ISBN 0-521-39063-X