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.[1]
For example the equation
<math>f(x,y)=y^2+x^2-x^3=0</math>
has an acnode at the origin, because it is equivalent to
<math>y^2 = x^2 (x-1)</math>
and <math>x^2(x-1)</math> is non-negative only when <math>x</math> ≥ 1 or <math>x = 0</math>. Thus, over the real numbers the equation has no solutions for <math>x < 1</math> 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 <math>\partial f\over \partial x</math> and <math>\partial f\over \partial y</math> 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.
