# Putnam 2003, prob B1 solved the hard way

1. Oct 20, 2004

### fourier jr

we're doing tensor products in my algebra class. here's an example the prof gave us.

problem: Can the polynomial $$1+xy+x^2y^2$$ be written in the form $$a(x)b(y)+c(x)d(y)$$ for polynomials a, b, c, d?

sol: $$\mathbb{R}^3 \otimes \mathbb{R}^3 \cong \mathbb{R}^3 \cong \mathcal{P}_2(x) \otimes \mathcal{P}_2(y) \cong \mathcal{P} \cong$$ polynomials in x & y whose degree is no greater than 2.

Let $$\Phi: \mathcal{P} \rightarrow \mathbb{R}^3$$. (which is an isomorphism) Then $$\Phi(1+xy+x^2y^2) = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) = I_3$$

$$\Phi(a(x)b(y)) =$$ a matrix with rank 1

$$\Phi(c(x)d(y)) =$$ a matrix with rank 1

The matrix $$I_3$$ has rank 3, but the sum of ranks is subadditive, ie. the sum of ranks of $$\Phi(a(x)b(y))$$ & $$\Phi(c(x)d(y))$$ can be 0, 1, 2 but not 3, so it is impossible.

Last edited: Oct 20, 2004