# Polynomial Decomposition

Decide if there are polynomials $A,B,C,D,E,F$ over $\mathbb{Q}$ such that

$1 + xy + x^2y^2 + x^3y^3 = A(x)B(y) + C(x)D(y) + E(x)F(y)$.

1. Let us prove that, more generally, $p(x,y)=\sum_{i=0}^n x^iy^i \neq \sum_{i=1}^n f_i(x)g_i(y)$ for $f_i,g_i\in k[t]$, with $k$ a field with enough elements.
Suppose such $f_i,g_i$ existed. Pick $n+1$ different points $x_0, \dots, x_n$ in $k$ and evaluate $p(x,y)$ in $x=x_j$. We obtain for each $j$ that $\sum_{i=0}^n x_j^iy^i$ is a $k$-linear combination of the polynomials $g_1,\dots, g_n$. However, the polynomials $\sum_{i=0}^n x_j^iy^i$ are linearly independent: if we write down their coordinates in the usual basis $\{1,y,\dots, y^n\}$ as rows of a square matrix we obtain the Vandermonde matrix associated to the tuple $(x_0,\dots, x_n)$, which is invertible. Therefore we have shown that we can write down $n+1$ linearly independent elements as linear combinations of $n$ polynomials, which is a contradiction.