This question is from the Putnam Competition, 2003. It asks a fairly straightforward question:

Problem.

Do there exist polynomials

$$ a(x), b(x), c(y), d(y) $$

such that

$$ 1+xy+x^2y^2=a(x)c(y)+b(x)d(y)? $$

Solution

This is a rather interesting question because it has completely no restrictions on the polynomials. In fact, one fails (at least most people did), if we use the standard comparison method, by taking the general expressions. We do not know clearly on the degrees of the polynomial. Hence, the classic coefficient-comparison effect does not really apply here.

Instead, we may approach via substition: taking

$$ y=0, y=1, y=-1 $$

Consecutively in the identity above yields the identities

$$ 1=c(0)a(x)+d(0)b(x) \\ 1+x+x^2=c(1)a(x)+d(1)b(y)\\1-x+x^2=c(2)a(x)+d(2)b(x) $$

Which implies that the polynomials

$$ 1, 1+x+x^2, 1-x+x^2 $$

are linear combinations of a(x) and b(x). However, considering the Wronskian

$$ W(1,1+x+x^2,1-x+x^2)= \begin{vmatrix} 1 & 1+x+x^2 & 1-x+x^2 \\ 0 & 1+2x &-1+2x \\0 & 2& 2\end{vmatrix} = 4 >0 $$

and hence they are linearly independent. Contradiction. Hence, there are no such polynomials.