Let $A$ be a commutative ring, partially ordered by $\le$. Consider the following proposition:
Proposition 1. Let $a, b, c$ be in $A$ and assume $a \ge 0$ and $c \ge 0$. If $a n^2 + 2 b n m + c m^2 \ge 0$ for all integers $n$ and $m$, then $b^2 - a c \le 0$.
As is well known, this is true for $A = \mathbb{R}$. However, it is also true for many other kinds of rings:
- The product ring $\prod_{i \in I} A_i$ satisfies proposition 1 if each factor $A_i$ satisfies proposition 1.
- Any subring of $A$ satisfies proposition 1 if $A$ satisfies proposition 1.
- Any $\aleph_1$-filtered colimit of rings satisfying proposition 1 also satisfies proposition 1.
Thus, for instance, the ring of continuous, smooth, real-analytic, etc. functions satisfies proposition 1. Consider also:
Proposition 2. Let $a, b, c$ be in $A$ and assume $a \ge 0$ and $c \ge 0$. If $a x^2 + 2 b x y + c y^2 \ge 0$ for all $x$ and $y$ in $A$, then $b^2 - a c \le 0$.
Of course, any ring that satisfies proposition 1 also satisfies proposition 2. I can prove proposition 2 if I assume either $a$ or $c$ divides $b$. Indeed, $$0 \le a (a x^2 - 2 b x + c) = a^2 x^2 - 2 a b x + a c = (a x - b)^2 - (b^2 - a c)$$ so if $a x = b$ then $b^2 - a c \le 0$ as claimed. This uses only the axioms of partially ordered rings. In the converse direction, if $a z = b$, then $$a x^2 + 2 b x y + c y^2 = a x^2 + 2 a x y z + c y^2 = a (x + y z)^2 - (a z^2 - c) y^2$$ so if $a z^2 - c \le 0$, and we assume all squares in $A$ are $\ge 0$, then $a x^2 + 2 b x y + c y^2 \ge 0$. Thus, a strong version of proposition 2 is true in any linearly ordered field.
Question. Is there a reasonably general class of partially ordered rings for which proposition 2 can be proved?
By "reasonably general class", I mean including at least rings of continuous functions, and axiomatised by a geometric theory in the language of partially ordered rings, i.e. a list of universally quantified implications of the form $p \to q$ where $p$ and $q$ are (possibly infinitary) disjunctions of existentially quantified finitary conjunctions of atomic predicates.
Essentially, I am looking for a proof that can handle the case where $a$ and $c$ are not invertible (or maybe even zero-divisors). Considering the case of rings of continuous functions, it seems to me that the crux of the matter must be the divisibility of $b$ by $a$ or $c$, but I have not been able to come up with reasonable axioms that imply this. (Perhaps it follows from the axioms of real closed rings?)