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".[1]
An acnode at the origin (curve described in text)
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
\( y^2 = x^2 (x-1) \)
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 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.
See also
Singular point of a curve
Crunode
Cusp
Tacnode
References
Hazewinkel, M. (2001) [1994], "Acnode", Encyclopedia of Mathematics, EMS Press
Porteous, Ian (1994). Geometric Differentiation. Cambridge University Press. ISBN 978-0-521-39063-7.
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