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 {\displaystyle 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 ) {\displaystyle y^{2}=x^{2}(x-1)}

and x 2 ( x − 1 ) {\displaystyle x^{2}(x-1)} is non-negative only when x {\displaystyle x} ≥ 1 or x = 0 {\displaystyle x=0} . Thus, over the real numbers the equation has no solutions for x < 1 {\displaystyle 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 ∂ f ∂ x {\displaystyle \partial f \over \partial x} and ∂ f ∂ y {\displaystyle \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.