Quotient of Homogeneous Functions

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:

The result follows from the definition.