Acnode

File:Isolated-point.svg

An acnode at the origin (curve described in text)

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 $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ 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

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