decomposable curve

An algebraic curve

$$f(x,y)=\mathrm{\hspace{0.33em}0}$$

is decomposable, if the polynomial  $f(x,y)$  is in  $ℝ[x,y]$; that is, if there are polynomials  $g(x,y)$  and  $h(x,y)$  with positive degree in  $ℝ[x,y]$  such that

$$f(x,y)=g(x,y)h(x,y).$$

Example.  The quadratic curve

${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=\mathrm{\hspace{0.33em}0}$ (1)

is decomposable, since the equation may be written

$$\left(\frac{x}{a}+\frac{y}{b}\right)\left(\frac{x}{a}-\frac{y}{b}\right)=\mathrm{\hspace{0.33em}0}$$

or equivalently

$$\frac{x}{a}+\frac{y}{b}=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\vee \mathit{\hspace{1em}}\frac{x}{a}-\frac{y}{b}=\mathrm{\hspace{0.33em}0}.$$

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

Analogically, one can say that an algebraic surface

$$g(x,y,z)=\mathrm{\hspace{0.33em}0}$$

is decomposable, e.g.  ${(x+y+z)}^2-1=0$  which consists of two parallel planes.

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