Acnode
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Anacnodeis anisolated pointin the solution set of apolynomial equationin two real variables. Equivalent terms areisolated pointandhermit point.[1]
For example the equation
has an acnode at the origin, because it is equivalent to
andis non-negative only when≥ 1 or.Thus, over therealnumbers the equation has no solutions forexcept 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 derivativesandvanish. Further theHessian matrixof second derivatives will bepositive definiteornegative definite,since the function must have a local minimum or a local maximum at the singularity.
See also
[edit]References
[edit]- ^Hazewinkel, M. (2001) [1994],"Acnode",Encyclopedia of Mathematics,EMS Press
- Porteous, Ian (1994).Geometric Differentiation.Cambridge University Press.ISBN978-0-521-39063-7.