decomposable curve

An algebraic curve

f(x,y)= 0

is decomposablePlanetmathPlanetmath, if the polynomialMathworldPlanetmathf(x,y)  is in  [x,y]; that is, if there are polynomials  g(x,y)  and  h(x,y)  with positive degree in  [x,y]  such that


Example.  The quadratic curve

x2a2-y2b2= 0 (1)

is decomposable, since the equation may be written

(xa+yb)(xa-yb)= 0

or equivalently

xa+yb= 0xa-yb= 0.

Thus the curve (1) consists of two intersecting lines.

Analogically, one can say that an algebraic surface

g(x,y,z)= 0

is decomposable, e.g.  (x+y+z)2-1=0  which consists of two parallel planesMathworldPlanetmath.

Title decomposable curve
Canonical name DecomposableCurve
Date of creation 2013-03-22 19:19:38
Last modified on 2013-03-22 19:19:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 08A40
Classification msc 26A09
Related topic Hyperbola2
Defines decomposable
Defines decomposable surface