Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows:

Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$ such that every $f\in \mathbb R[x,y]_+$ can be represented as $$ f=\sum_{i=1}^n p_i^2\cdot h_i,\ h_i\in M,\ p_i\in\mathbb R[x,y]\ ? $$ More concretely, can we take $M=\mathbb Q[x,y]_+\ ?$

Recall that a *quadratic module* in a commutative ring $A$ with $1\in A$ is a subset $Q\subseteq A$ such that

$(i)\ 1\in Q,\ -1\notin Q$, $(ii)\ Q+Q\subseteq Q$, $(iii)\ p^2\cdot Q\subseteq Q,\ \forall p\in A.$

The problem can be now stated as follows:

Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

The smallest quadratic module in $\mathbb R[x_1,\dots,x_d]$ is the set $\sum^2\mathbb R[x_1,\dots,x_d]$ of finite sums of squares. We have $\mathbb R[x_1,\dots,x_d]_+=\sum^2\mathbb R[x_1,\dots,x_d]$ iff $d=1$ (Hilberts theorem). It is known that $\mathbb R[x_1,\dots,x_d]_+$ is not finitely generated for $d\geq 2.$ One can also easily show that $\mathbb R[x_1,\dots,x_d]_+$ is not countably generated for $d\geq 3.$