An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are "
isolated point or hermit point".
For example the equation
:
has an acnode at the origin, because it is equivalent to
:
and
is non-negative only when
≥ 1 or
. Thus, over the ''real'' numbers the equation has no solutions for
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
and
vanish. Further the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of second derivatives will be
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
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
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic cur ...
*
Crunode
In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ''ordinary double point''.
For a ...
*
Cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
*
Tacnode
References
*
Curves
Algebraic curves
Singularity theory
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es:Punto singular de una curva#Acnodos