acnode

{{Short description|Isolated point in the solution set of a polynomial equation in two real variables}}

Image:Isolated-point.svg

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.{{SpringerEOM| title=Acnode | id=Acnode | oldid=15498 | first=M. | last=Hazewinkel }}

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 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 $\backslash partial\; f\backslash over\; \backslash partial\; x$ and $\backslash partial\; f\backslash over\; \backslash partial\; y$ 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.