Proposition

If $h \in k[x_1, \cdots, x_n]$ has total degree $N$ and if $\displaystyle F(t_1, \cdots, t_n) = \big(\frac{f_1(t_1, \cdots, t_m)}{g_1(t_1, \cdots, t_m)}, \cdots, \frac{f_n(t_1, \cdots, t_m)}{g_n(t_1, \cdots, t_m)}\big)$, show that $(g_1 \cdots g_n)^N(h \circ F)$ is a polynomial in $k[t_1, \cdots, t_m]$.

Solution

If $h$ is not, then $(g_1 \cdots g_n)^N(h \circ F) = \sum (g_1 \cdots g_n)^N(h_i \circ F)$ where $h_i$’s are the monomials of $h$. Therefore, it suffices to show this for the case that $h$ is a monomial without a coefficient. Suppose $h = F_1^{k_1} \cdots F_n^{k_n}$ where each $k_i \geq 0$ and $\sum k_i = N$.

Then