64th Putnam 2003

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)?

Answer

No.

Solution

Suppose there are such polyonomials. Write c_n for c(n), d_n 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
jscholes@kalva.demon.co.uk
8 Dec 2003
Last corrected/updated 8 Dec 03