I am not working in the field of algorithmic algebraic geometry - yet, for my current work, I need some results from it.

More specifically, what is the state-of-the-art when it comes to solving (whatever "solving" means in this case) system of polynomials of fields that are not algebraically closed, whose ideal has dimension $>0$?

Could you recommend a survey paper that summarizes what has been achieved so far?

For the case of $0$-dimensional ideals, there seems to exist many heavily cited papers, like "Solving Zero-dimensional Algebraic Systems" by D. Lazard, which seem mostly to be concerned with finding ways of to display the system of polynomials in a nice way (e.g. triangularly). Are these articles already superseded, or does it make sense to read them?

**Edit:** In particular, I'm interested in the field $\mathbb{R}$, since most of my example will come from here (but $\mathbb{Q}$ might be also useful; and perhaps even the ring $\mathbb{Z}$; I don't yet know where the results I will get for $\mathbb{R}$ will take me).

Also worth making more precise: In the case of positive dimension of the ideal, I'm interested in methods that tell me, if I project to whole, infinite solution space down to a single variable and I'm interested in, in what set this variable lies. More formally, if $V(f_1,\ldots,f_s)\subseteq F^n$ is my solution variety, with $f_i \in F[x_1,\ldots,x_n]$, and I'm interested in some specific variable, say $n_0$, what methods are there that describe $\mathop{\rm proj}_{n_0}(V(f_1,\ldots,f_s))$?