Let $f(X_1,\ldots,X_n)$ be a polynomial with integer coefficients, in $n$ variables. For every $\lambda\in\mathbb{C}$, i have
$$f(\lambda X_1,\ldots,\lambda X_n)=\lambda^kf(X_1,\ldots,X_n)$$.
Can I conclude that $f$ is homogeneous of degree $k$? Why?
My definition of polynomial homogeneous of degree $k$ is: a polynomial whose all nonzero terms have the same degree $k$.
Best Answer
You already know $f$ is a polynomial in $n$ variables so you can write it in the general form $$f(X_1,...,X_n)=\sum_{j_1,j_2,...j_n=0}^nc_kX_1^{j_1}X_2^{j_2}...X_n^{j_n}$$ for some $c_i\in \mathbb{Z}$, $j_i=0,...,n$, $i=0,...,n$. Calculate this with $\lambda X_i$ in the argument
$$f(\lambda X_1,...,\lambda X_n)=\sum_{j_1,j_2,...j_n=0}^nc_k(\lambda X_1)^{j_1}(\lambda X_2)^{j_2}...(\lambda X_n)^{j_n}$$ $$=\sum_{j_1,j_2,...j_n=0}^nc_k\lambda^{j_1+...+j_n} X_1^{j_1}X_2^{j_2}...X_n^{j_n}$$
Comparing this with $$\lambda^kf(X_1,...,X_n)=\lambda^k\sum_{j_1,j_2,...j_n=0}^nc_kX_1^{j_1}X_2^{j_2}...X_n^{j_n}$$ $$=\sum_{j_1,j_2,...j_n=0}^nc_k\lambda^k X_1^{j_1}X_2^{j_2}...X_n^{j_n}$$ for every $\lambda \in \mathbb{C}$ we can see that $$j_1+...+j_n=k$$ But this means precisely that $f$ is homogeneous of degree $k$