Anacnode is anisolated point in the solution set of apolynomial equation in two real variables. Equivalent terms areisolated point andhermit point.^[1]

For example the equation

${\displaystyle f(x,y)=y^2+x^2-x^3=0}$

has an acnode at the origin, because it is equivalent to

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

and${\displaystyle x^2(x-1)}$ is non-negative only when${\displaystyle x}$ ≥ 1 or${\displaystyle x=0}$. Thus, over thereal numbers the equation has no solutions for 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, orsingularity, of the defining polynomial function, in the sense that both partial derivatives$\frac{{\displaystyle \mathrm{\partial }f}}{\mathrm{\partial }x}$ and$\frac{{\displaystyle \mathrm{\partial }f}}{\mathrm{\partial }y}$ vanish. Further theHessian matrix of second derivatives will bepositive definite ornegative definite, since the function must have a local minimum or a local maximum at the singularity.

• Singular point of a curve
• Crunode
• Cusp
• Tacnode

• Porteous, Ian (1994).Geometric Differentiation.Cambridge University Press. p. 47.ISBN 978-0-521-39063-7.

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