# Acnode

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

An '''acnode''' is an [isolated point](/source/isolated_point) in the solution set of a [polynomial equation](/source/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 |author-link=Michiel 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](/source/singularity_theory), 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](/source/Hessian_matrix) of second derivatives will be [positive definite](/source/Positive-definite_matrix) or [negative definite](/source/Negative-definite_matrix), since the function must have a local minimum or a local maximum at the singularity.

==See also==
*[Singular point of a curve](/source/Singular_point_of_a_curve)
*[Crunode](/source/Crunode)
*[Cusp](/source/Cusp_(singularity))
*[Tacnode](/source/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](/source/Cambridge_University_Press) |isbn=978-0-521-39063-7 |page=47}}

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[es:Punto singular de una curva#Acnodos](/source/es%3APunto_singular_de_una_curva)

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Adapted from the Wikipedia article [Acnode](https://en.wikipedia.org/wiki/Acnode) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Acnode?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.