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[parent] homogeneous equation (Topic)

The homogeneous equation

$\displaystyle f(x,\,y) = 0,$
where the left hand side is a homogeneous polynomial of degree $ r$ in $ x$ and $ y$, determines the ratio $ x/y$ between the indeterminates. One can be persuaded of this by dividing both sides of the equation by $ y^r$. Then the left side depends only on $ x/y$ (which may be denoted e.g. by $ t$).

Examples

  • The equation $ 5x+8y = 0$ determines that $ x/y = -\frac{8}{5}$.
  • The equation $ x^2-7xy+10y^2 = 0$ determines that $ x/y = 2$ or $ x/y = 5$ (these values are obtained by first dividing both sides of the equation by $ y^2$ and then solving the equation $ (x/y)^2-7(x/y)+10 = 0$).
  • The equation $ 2x^3-x^2y-6xy^2+3y^3 = 0$ determines that $ x/y = \frac{1}{2}$ or $ x/y = \pm\sqrt{3}$ (first divide the equation by $ y^3$ and then solve $ 2(x/y)^3-(x/y)^2-6(x/y)+3 = 0$).



"homogeneous equation" is owned by pahio.
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See Also: variation, homogeneous polynomial, equation, regular decagon inscribed in circle

Keywords:  proportional

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Cross-references: divide, equation, indeterminates, ratio, degree, homogeneous polynomial
There are 6 references to this entry.

This is version 4 of homogeneous equation, born on 2005-05-07, modified 2006-02-25.
Object id is 7022, canonical name is HomogeneousEquation.
Accessed 4086 times total.

Classification:
AMS MSC: 00A99 (General :: General and miscellaneous specific topics :: Miscellaneous topics)
 26B35 (Real functions :: Functions of several variables :: Special properties of functions of several variables, Hölder conditions, etc.)
 26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.)
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