Let be a polynomial such that for all . Prove that for some polynomials .
Note that must have even degree. We proceed by induction on the degree of . Note as a lemma that if and are sums of squares, then so is since (evidently) . We find this identity by rearranging and partially simplifying the factorization .
The base case is trivial; if and if .
For the inductive step, suppose the result holds for all polynomials of degree and let have degree . Suppose has a real root . Now is concave up on a sufficiently small neighborhood about , so that . In particular, is a root of of multiplicity at least 2. Say . Now is a sum of two squares, and is as well, so that by the lemma is a sum of two squares. Suppose instead that has no real roots. Instead, we have a complex root . Since conjugation is a ring homomorphism, is also a root of . Letting , we see that is a factor of ; say . Again, is a sum of two squares.