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Let $k$ be a field.

Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$.

If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!

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Yes. More generally:

Let $A = \bigoplus_{i\in\mathbb{Z}} A_i$ be a graded domain und $f\in A\setminus \{0\}$ homogeneous. If $f$ factors in $A$ as $f=gh$ then $g,h$ are homogeneous.

Proof: Since we are in a domain and $f\neq 0$, the factors $g,h$ are non-zero as well. Let $g$ have non-zero component of lowest degree $d_{\min}$ and of highest degree $d_{\max}$. Similarly for $h$ (with degree symbol $e$ in place of $d$). Then $f=gh$ has non-zero component of lowest degree $d_{\min}+ e_{\min}$ and of highest degree $d_{\max} + e_{\max}$ (here we use that $A$ is a domain!). But since $f$ is homogeneous, lowest and highest degree of $f$ agree, i.e. $d_{\min}+ e_{\min} = d_{\max}+ e_{\max}$. From

$$0 = (\;d_{\max} – d_{\min}\;) + (\;e_{\max} – e_{\min}\;)$$

and the non-negativity of the summands we conclude $d_{\min} = d_{\max}$ and $e_{\min} = e_{\max}$, i.e., $g$ and $h$ are homogeneous. **qed**

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