# decomposable curve

 $f(x,\,y)\;=\;0$

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

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

 $\displaystyle\frac{x^{2}}{a^{2}}\!-\!\frac{y^{2}}{b^{2}}\;=\;0$ (1)

is decomposable, since the equation may be written

 $\left(\frac{x}{a}\!+\!\frac{y}{b}\right)\!\left(\frac{x}{a}\!-\!\frac{y}{b}% \right)\;=\;0$

or equivalently

 $\frac{x}{a}\!+\!\frac{y}{b}\;=\;0\quad\lor\quad\frac{x}{a}\!-\!\frac{y}{b}\;=% \;0.$

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 planes.

Title decomposable curve DecomposableCurve 2013-03-22 19:19:38 2013-03-22 19:19:38 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 08A40 msc 26A09 Hyperbola2 decomposable decomposable surface