Acnode Explained

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 toy ^{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

• Book: Porteous, Ian . Geometric Differentiation . registration . 1994 . . 978-0-521-39063-7 .