Do their exist polynomials a(x), b(x), c(y), d(y) such that 1 + xy + x2y2 ≡ a(x)c(y) + b(x)d(y)?
Suppose there are such polyonomials. Write cn for c(n), dn for d(n). Then 1 = c0a(x) + d0b(x), 1 + x + x2 = c1a(x) + d1b(x), 1 - x + x2 = c-1a(x) + d-1b(x). So we have:
1 = c0a(x) + d0b(x)
x = (-½c0+½c1-½c-1)a(x) + (-½d0+½d1-½d-1)b(x)
x2 = (-c0+½c1+½c-1)a(x) + (-d0+½d1+½d-1)b(x)
So we have three linearly independent elements (the polynomials 1, x, x2) in a subspace of dimension 2 (the vector space spanned by a(x) and b(x)). Contradiction.
64th Putnam 2003
© John Scholes
8 Dec 2003
Last corrected/updated 8 Dec 03