I’ve been doing some more online reading on the Positivstellensatz. I had blithely assumed that the polynomials that are non-negative for all real values were given by sums of squares of polynomials, but this is false. What is true is that a polynomial is non-negative if and only if it can be written as the sums of squares of rational functions, but this is a nontrivial result. In fact, showing this was Hilbert’s seventeenth problem. (You can derive it from the Positivstellensatz.)

If you were to try to axiomatize the idea of positivity inside a commutative ring in such a way that the same set of axioms cover both positive real numbers and non-negative polynomials, you would include axioms such as the sum and product of two positive elements are positive, and that squares are always positive. Hilbert’s seventeenth problem shows that you need an additional axiom: if a and ab are positive, then so is b.