Acnode

An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point.

For example the equation
:$f(x,y)=y^2+x^2-x^3=0$
has an acnode at the origin, because it is equivalent to
:$y^2 = x^2 (x-1)$
and $x^2(x-1)$ is non-negative only when $x$ ≥ 1 or $x = 0$. 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. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point.

An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives $\partial f\over \partial x$ and $\partial f\over \partial y$ vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite, since the function must have a local minimum or a local maximum at the singularity.

See also

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

References

*

Curves
Algebraic curves
Singularity theory

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