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If $f_{1}(x,y)$ and $f_{2}(x,y)$ are two homogeneous polynomials then prove that :

1

$frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$ is homogeneous.

2

Can i replace the denominator part by anyother homogeneous polynomial? Will it be homogeneous then ?
Say
$f_{3}(x,y)$ be another homogeneous polynomials then what about $frac{f_{1}(x,y)+f_{2}(x,y)}{f_{3}(x,y)}$ ?

A function $h( mathbf x)$, where $mathbf x=(x_1,x_2,...,x_n)$ is said to be homogeneous of degree $k$ if

$$h( lambda mathbf x)=lambda^k h( mathbf x)$$

for all $lambda in Bbb R.$

Even though $f_1(x,y)$ and $f_2(x,y)$ are both homogeneous, their sum will, in general, not be homogeneous, because their degree of homogeneity need not be the same

$$f_1(lambda x, lambda y)+f_2(lambda x,lambda y)=lambda^{k_1} f_1(x,y)+lambda^{k_2} f_2(x,y) neq lambda^k[f_1(x,y)+f_2(x,y)].$$

Of course, if it happens that their degree of homogeneity is the same, then $k_1=k_2=k$ and

$$f_1(lambda x, lambda y)+f_2(lambda x,lambda y)=lambda^{k} f_1(x,y)+lambda^{k} f_2(x,y)=lambda^{k} [f_1(x,y)+f_2(x,y)].$$

Therefore,

$frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$ is homogeneous if the degree of homogeneity of $f_{i}(x,y)$ is the same for all $i$.

And you can replace the denominator part by any other homogeneous polynomial. In this case, the degree of homogeneity does not matter, because (assuming again that
$f_1(x,y)$ and $f_2(x,y)$ are both homogeneous), we have that

$$frac{f_1(lambda x, lambda y)}{f_2(lambda x,lambda y)}=frac{lambda^{k_1} f_1(x,y)}{lambda^{k_2} f_2(x,y)}=lambda^{k_1-k_2}frac{f_1(x,y)}{f_2(x,y)}.$$