Suppose we have $n $ variables $x_{1\leq i \leq n }$. Consider the vector space of degree-$d$ homogeneous polynomials of $x_i$ over the field $\mathbb{C}$. The dimension of this space is $D_1 = (d+n -1)!/(n-1)!/d!$.

Consider the special set of polynomials (which we shall refer to as decomposable polynomials) which can be written as the product of $d$ linear forms

$$ \Omega = \prod_{j=1}^d \left( \sum_{i=1}^n a_{ij }x_i \right ) .$$

It is not hard to see that the degrees of freedom of $\Omega$ is $D_2 = d(n-1) +1 $.

Now the question is, is it always possible to write an arbitrary $d$-degree homogeneous polynomial $F$ as **the sum** of $M = \lceil D_1/D_2 \rceil$ decomposable polynomials? Here $\lceil \cdot \rceil $ is the ceiling function.

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