Acnode

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An acnode at the origin (curve described in text)

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]

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

f(x,y)=y^2+x^2-x^3=0\;

has an acnode at the origin of \mathbb{R}^2, because it is equivalent to

y^2 = x^2 (x-1)

and x^2(x-1) is non-negative when x ≥ 1 and 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 \partial f\over \partial x and \partial f\over \partial y vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite. Hence the function has a local minimum or a local maximum.

See also[edit]

References[edit]

  1. ^ Hazewinkel, M. (2001), "Acnode", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4