Serre proved that every polynomial of with odd (with a small number of exceptions when ) is a strict sum of three squares. The authors prove by using the same method, (apply Weil’s theorem for an appropriate curve) that for even all (but a finite number of polynomials when , all explicitly stated in the paper) polynomials of are of the form (we say that they are decomposable):
where satisfy the tight condition:
The exceptions are well behaved in the sense that it is easy to prove that for all of them over and over are decomposable. Thus, every polynomial in has a strict representation of the form:
It is also proved that for every even the only quadratic polynomials in three variables that represent strictly all (but a finite number) of polynomials of are
Observe that strict representations by the first and the last quadratic polynomials are trivial.