For a given polynomial $f(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$, define the height of $f$ as $H(f)$ as the maximum absolute value among its coefficients. We can also define the log height $h(f) = \log H(f)$.

If $f = f_1 \cdots f_r, f_j \in \mathbb{Z}[x_1, \cdots, x_n]$ and $d = \deg f$, then Gelfond's inequality states that $$\displaystyle \sum_{j=1}^r h(f_j) \leq h(f) + nd.$$

My question is the following: Suppose $F(x_0, \cdots, x_n) \in \mathbb{Z}[x_0, \cdots, x_n]$ is homogeneous with degree $d$ and is primitive (i.e., its coefficients are co-prime), which is irreducible over $\mathbb{Q}$ but is not absolutely irreducible. Let $K/\mathbb{Q}$ be a number field such that $F$ splits completely into absolutely irreducible factors $F_1, \cdots, F_r$. Then the $F_i$'s are necessarily conjugate. Further, for each $j, 1 \leq j \leq r$ we can write $$\displaystyle F_j = \sum \lambda_{i,j} F_{i,j}$$ where the $\lambda_{i,j}$'s are $\mathbb{Q}$-linearly independent and the $F_{i,j} \in \mathbb{Z}[x_0, \cdots, x_n]$ for all $i$. Indeed, by noting that the $F_j$'s are conjugate we see that the $F_{i,j}$'s don't depend on $j$. Let $G = F_{u,v}$ be such that $G$ has the smallest (projective) height among all $F_{i,j}$'s. Can we bound $H(G)$ or $h(G)$ in terms of $F$?