Apply Euler’s formula on a function which is the sum of two homogeneous functions.

Question

So we can use Euler's formula on homogeneous two variables functions of any degree and write the function in the form of its partial derivatives.
But can we still use the formula when the function is made of two homogeneous functions with different degrees?
For example if$f(x,y)=h(x,y)+k(x,y)$ such that $h$ is homogeneous of degree m and $k$ is homogeneous of degree n.
How do you apply Euler's formula here?

Answer 1

Sort of?

I will show with example:

According to euler's theorem:

$$ u(x,y) = \sum_{i+j=k}^{i \leq n,j \leq n} \beta_{i,j} x^i y^j$$

Then euler's theorem states that:

$$ ku = x\partial_x u + y\partial_y u$$

Then, suppose we have sum of two homogenous polynomial of degree $n$ and $m$:

$$ q(x,y) = r(x,y) + t(x,y)$$

Then I can say that: $$ q= x\partial_x ( \frac{r}{n}+\frac{t}{m}) +y \partial_y (\frac{r}{n}+\frac{t}{m})$$

By the fact that:

$$ nr(x,y) = x\partial_x r + y\partial_y r$$

Or,

$$ r = \frac{1}{n} (x\partial_x r +y \partial_y r)$$

We can write a similar substituion of $t$, from which we can write $q$ as:

$$ q= \frac{1}{n} (x\partial_x r +y \partial_y r) + \frac{1}{m} (x\partial_x t +y \partial_y t)$$

Summary: Adding two homogenous function of non equal degree results in a non homogenous polynomial leading to inapplicability of Euler's theorem but we can still express each polynomial in the sum as a homogenous polynomial via Euler's theorem

Note:

$$ \partial_i r = \frac{\partial}{\partial i} r$$