An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or 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

\partial f\over \partial x

and

\partial f\over \partial y

vanish. Further the

Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...

of second derivatives will be

positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...

or negative definite, since the function must have a local minimum or a local maximum at the singularity.

References

* Curves Algebraic curves Singularity theory {{algebraic-geometry-stub es:Punto singular de una curva#Acnodos

See also

References