Acnode

An acnode is an <a href="/facts/Isolated_point/d3EeWxuD">isolated point</a> in the solution set of a <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a> in two real variables. Equivalent terms are isolated point and hermit point.
For example the equation

f
        (
        x
        ,
        y
        )
        =
        
          y
          
            2
          
        
        +
        
          x
          
            2
          
        
        −
        
          x
          
            3
          
        
        =
        0
      
    
    {\displaystyle f(x,y)=y^{2}+x^{2}-x^{3}=0}

has an acnode at the origin, because it is equivalent to

y
          
            2
          
        
        =
        
          x
          
            2
          
        
        (
        x
        −
        1
        )
      
    
    {\displaystyle y^{2}=x^{2}(x-1)}

and 
 
 
 
 
 x
 
 2
 
 
 (
 x
 −
 1
 )
 
 
 {\displaystyle x^{2}(x-1)}
 
 is non-negative only when 
 
 
 
 x
 
 
 {\displaystyle x}
 
 ≥ 1 or 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
. Thus, over the real numbers the equation has no solutions for 
 
 
 
 x
 <
 1
 
 
 {\displaystyle 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. 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 <a href="/facts/Singularity_theory/MgLkyrhl">singularity</a>, of the defining polynomial function, in the sense that both partial derivatives 
 
 
 
 
 ∂
 f
 
 
 ∂
 x
 
 
 
 {\displaystyle \partial f \over \partial x}
 
 and 
 
 
 
 
 ∂
 f
 
 
 ∂
 y
 
 
 
 {\displaystyle \partial f \over \partial y}
 
 vanish. Further the <a href="/facts/Hessian_matrix/RguMNr3m">Hessian matrix</a> of second derivatives will be <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive definite</a> or <a href="/facts/Negative-definite_matrix/Hbr6AuPS">negative definite</a>, since the function must have a local minimum or a local maximum at the singularity.

Acnode open-in-new

Acnode