Quotient of Homogeneous Functions

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Theorem

Let $\map M {x, y}$ and $\map N {x, y}$ be homogeneous functions of the same degree.


Then:

$\dfrac {\map M {x, y} } {\map N {x, y} }$

is homogeneous of zero degree.


Proof

Let:

$\map Q {x, y} = \dfrac {\map M {x, y} } {\map N {x, y} }$

where $M$ and $N$ are homogeneous functions of degree $n$.


Let $t \in \R$.

Then:

\(\ds \map Q {t x, t y}\) \(=\) \(\ds \frac {\map M {t x, t y} } {\map N {t x, t y} }\)
\(\ds \) \(=\) \(\ds \frac {t^n \map M {x, y} } {t^n \map N {x, y} }\) as $M$ and $N$ are homogeneous of degree $n$
\(\ds \) \(=\) \(\ds t^0 \frac {\map M {x, y} } {\map N {x, y} }\)
\(\ds \) \(=\) \(\ds t^0 \map Q {x, y}\)
\(\ds \) \(=\) \(\ds \map Q {x, y}\)

The result follows from the definition.

$\blacksquare$


Sources