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).

Polynomial Decomposition

One thought on “Polynomial Decomposition

  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.

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