Acnode

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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 :$f\left(x,y\right)=y^2+x^2-x^3=0$ has an acnode at the origin, because it is equivalent to :$y^2 = x^2 \left(x-1\right)$ and $x^2\left(x-1\right)$ 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 (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessi ...
of second derivatives will be
positive definite In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
or
negative definite In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite bilinear form, negative-definite. See, in particular: * Negative-definite bilinear form * Negative-de ...
, since the function must have a local minimum or a local maximum at the singularity.