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{{Short description|Isolated point in the solution set of a polynomial equation in two real variables}}
[[Image:Isolated-point.svg|thumb|right|An acnode at the origin (curve described in text)]]

An '''acnode''' is an [[isolated point]] in the solution set of a [[polynomial equation]] in two real variables. Equivalent terms are '''isolated point ''' and '''hermit point'''.<ref>{{SpringerEOM| title=Acnode | id=Acnode | oldid=15498 | first=M. | last=Hazewinkel }}</ref>

For example the equation
:<math>f(x,y)=y^2+x^2-x^3=0</math>
has an acnode at the origin, because it is equivalent to
:<math>y^2 = x^2 (x-1)</math>
and <math>x^2(x-1)</math> is non-negative only when <math>x</math> ≥ 1 or <math>x = 0</math>. Thus, over the ''real'' numbers the equation has no solutions for <math>x < 1</math> 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 theory|singularity]], of the defining polynomial function, in the sense that both partial derivatives <math>\partial f\over \partial x</math> and <math>\partial f\over \partial y</math> vanish. Further the [[Hessian matrix]] of second derivatives will be [[Positive-definite matrix|positive definite]] or [[Negative-definite matrix|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 (singularity)|Cusp]]
*[[Tacnode]]

==References==
{{reflist}}
*{{cite book |last=Porteous |first=Ian |title=Geometric Differentiation |url=https://archive.org/details/geometricdiffere0000port |url-access=registration |year=1994 |publisher=[[Cambridge University Press]] |isbn=978-0-521-39063-7 |page=47}}

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[[Category:Curves]]
[[Category:Algebraic curves]]
[[Category:Singularity theory]]

{{algebraic-geometry-stub}}

[[es:Punto singular de una curva#Acnodos]]

Acnode - Revision history

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