{{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 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 |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, 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 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== {{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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