# Quotient of Homogeneous Functions

From ProofWiki

## 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 degree zero.

## 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:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle Q \left({tx, ty}\right)\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac {M \left({tx, ty}\right)} {N \left({tx, ty}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac {t^n M \left({x, y}\right)} {t^n N \left({x, y}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | as $M$ and $N$ are homogeneous of degree $n$ | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle t^0 \frac {M \left({x, y}\right)} {N \left({x, y}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle t^0 Q \left({x, y}\right)\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle Q \left({x, y}\right)\) | \(\displaystyle \) | \(\displaystyle \) |

The result follows from the definition.

$\blacksquare$