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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 Are these articles already superseded, or does it make sense to read them?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 $\mathop{\rm proj}_{n_0}(V(f_1,\ldots,f_s))$?

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    $\begingroup$ This question is way too broad to give any kind of meaningful answer. If you can be more specific about how this question is relevant to your work, that should help to give the question more focus. $\endgroup$ – RP_ May 27 at 19:50
  • $\begingroup$ In particular, "solving" might be interpreted in various communities as either finding a single solution or finding all solutions. This gets murkier for positive-dimensional solution sets: how should one represent "all solutions"? $\endgroup$ – tim May 28 at 16:59
  • $\begingroup$ @RP_ hm... I don't think it would be that helpful to describe how solving polynomial systems arise in my work, since there are no restrictions on the structure of polynomials systems, so I'm still left with the "how can I solve general polynomials systems over the reals". more specifically, I'm investigating certain classes of neural nets and some metric properties of them, that lead to the questions whether some polynomial systems have solutions. $\endgroup$ – user43263 May 28 at 18:11
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    $\begingroup$ That is not what you asked: you asked about "fields that are not algebraically closed." Now it turns out you are only interested in the reals. Better edit your question. $\endgroup$ – RP_ May 28 at 18:48
  • $\begingroup$ @RP_ Well, I'm not exclusively interested in the reals, but it does consist in the most important case - see my edit. $\endgroup$ – user43263 May 28 at 19:00
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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.

For dimension $1$ specifically, Poonen also has a set of lecture notes on rational points on curves, 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...

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    $\begingroup$ It might be good to highlight the difference that, for the reals, the existence of an algorithm has been known for a long time, since work (I think) of Tarski, and it's "just" a question of improving the speed to be practical. For the rationals, the problem is completely open even in some of the simplest cases and one can easily organize multiple large conferences on non-overlapping aspects of the problem. $\endgroup$ – Will Sawin May 28 at 21:35
  • $\begingroup$ @WillSawin Yes absolutely, I kind of assumed that had already been mentioned somewhere... $\endgroup$ – RP_ May 28 at 21:59
  • $\begingroup$ Thanks, I wasn't aware of Poonen's book. After doing some reading I realized, given how comprehensive the theory is, that it is absolutely necessary to outline my problem in as much detail as possible. I have written a follow-uip question in that regard (mathoverflow.net/questions/361642/… ) , since giving all that detail it seemed outside the scope of this question. $\endgroup$ – user43263 May 29 at 8:32
  • $\begingroup$ @WillSawin Just for fun, could you link one paper (perhaps a survey paper, if there is one) that describes some unknown things in case of the rationals? $\endgroup$ – user43263 May 29 at 13:07
  • $\begingroup$ @user43263 Here is a list of open problems concerning K3 surfaces, which are sometimes advertised as being the frontier of rational points research in dimension $2$: sites.math.washington.edu/~bviray/openproblems.pdf Although this is an arguable point, since there are plenty of questions left about rational surfaces and generally about surfaces of Kodaira dimension $-1$. $\endgroup$ – RP_ May 29 at 17:13
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For the real field:

MR2830310 Sottile, Frank Real solutions to equations from geometry. University Lecture Series, 57. American Mathematical Society, Providence, RI, 2011.

MR2275625 Mikhalkin, Grigory Tropical geometry and its applications. International Congress of Mathematicians. Vol. II, 827–852, Eur. Math. Soc., Zürich, 2006.

MR1108621 Khovanskiĭ, A. G. Fewnomials. American Mathematical Society, Providence, RI, 1991.

MR1659509 Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry. Springer-Verlag, Berlin, 1998.

For other fields:

MR2247966 Vakil, Ravi Schubert induction. Ann. of Math. (2) 164 (2006), no. 2, 489–512.

Also:

MR1925796 Sturmfels, Bernd Solving systems of polynomial equations. CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002.

where the real field is also discussed.

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  • $\begingroup$ Thank you very much for all these references, in particular Sturmfels' book seems to contain useful theorems for me. $\endgroup$ – user43263 May 29 at 8:30

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