n.
[L. acus needle + E. node.]
(Geom.) An isolated point not upon a curve, but whose coördinates satisfy the equation of the curve so that it is considered as belonging to the curve.
| Dictionary: Ac·node |
[L. acus needle + E. node.]
(Geom.) An isolated point not upon a curve, but whose coördinates satisfy the equation of the curve so that it is considered as belonging to the curve.
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| Wikipedia: Acnode |
An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term. [1] [2]
Acnodes commonly occur when studying algebraic curves over fields which are not algebraically closed, defined as the zero set of a polynomial of two variables. For example the equation

has an acnode at the origin of
, because it is equivalent to
and x2 + x3 is positive for x > − 1, except when 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.
An acnode is a singularity of the function, where both partial derivatives
and
vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite. Hence the function has a local minimum or local maximum.
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| isolated | |
| isolated point (mathematics) | |
| conjugate |
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![]() | Dictionary. Webster 1913 Dictionary edited by Patrick J. Cassidy Read more | |
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