Decide if there are polynomials over such that

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# Polynomial Decomposition

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Decide if there are polynomials over such that

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Let us prove that, more generally, for , with a field with enough elements.

Suppose such existed. Pick different points in and evaluate in . We obtain for each that is a -linear combination of the polynomials . However, the polynomials are linearly independent: if we write down their coordinates in the usual basis as rows of a square matrix we obtain the Vandermonde matrix associated to the tuple , which is invertible. Therefore we have shown that we can write down linearly independent elements as linear combinations of polynomials, which is a contradiction.

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