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 Are these articles already superseded, or does it make sense to read them?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 $\mathop{\rm proj}\_{n\_0}(V(f\_1,\ldots,f\_s))$?



What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed?
For the reals, I particularly like the book by Sturmfels mentioned by Alexandre Eremenko. For the rational numbers, you can hardly do better than Bjorn Poonen's book *Rational Points on Varieties*, which is available for browsing via [his homepage](http://www-math.mit.edu/~poonen/). 


For dimension $1$ specifically, Poonen also has a set of [lecture notes on rational points on curves](https://math.mit.edu/~poonen/papers/curves.pdf), although I always have trouble finding it. Moreover he has several expository articles (listed as such on his page) dealing with rational points on curves.


Restricted to the case of the field of rational numbers and dimension $1$ alone, this is a huge question. Restricting only to the field of rational numbers makes it even huger. Dropping any restrictions on the field entirely makes it well-nigh impossible to answer in full geberality...


