Quotient of Homogeneous Functions

Theorem
Let $M \left({x, y}\right)$ and $N \left({x, y}\right)$ be homogeneous functions of the same degree.

Then:
 * $\dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

is homogeneous of zero degree.

Proof
Let:
 * $Q \left({x, y}\right) = \dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

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

Let $t \in \R$. Then:

The result follows from the definition.