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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 $F$?

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I think the answer is no, at least in the currently stated form. For instance, consider $x^2 + y^2 = (x+ iy)(x-iy)$. Now choose a matrix in $SL_2(\bf Z)$, i.e. choose $a,b,c,d$ integers such that $ad - bc = 1$. Let $x' = ax + by$ and $y' = cx + dy$ be linear forms. Then $x = dx' - by'$ and $y = -cx' + ay'$. So we get $$ x^2 + y^2 = \big( dx' - by' + i(-cx' + ay') \big) \big( dx' - by' - i(-cx' + ay') \big) = \big( (d - ic)x'+ (a-ib) y' \big) \big( (d+ic) x' - (a+ib) y' \big) $$ And now $x'$ and $y'$ have large height, if $a,b,c,d$ are large enough. So unless you have a better way of pinning down (a choice of) $F_{i,j}$, I don't think you can get an upper bound on their heights. (Maybe there is a bound if you take the minimum over all choices?)

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  • $\begingroup$ I believe you are right. I think the question is more reasonable if one supposes some sort of structure on the $F_{i,j}$'s. One way to think about it is to consider an integral basis $c_1, ..., c_d$ of the ring of integers in $\mathcal{O}_K$, then take a dual basis $\omega_1, \cdots, \omega_d$. Then one can write $F_j = \sum_{i=1}^d \lambda_i \operatorname{Tr}(F_j \omega_i)$. One can hopefully get a bound in terms of $F_j$ and the basis. $\endgroup$ – Stanley Yao Xiao Mar 16 '15 at 16:48

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