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In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial equation?

Failing that, has anyone written down a detailed description of such an algorithm? It may be that one (ahem) obstruction to the existence of such software is that the standard accounts of the Brauer–Manin obstruction are written in the language of modern algebraic geometry, which is unfamiliar to many people who might otherwise have the right skills to write the software. To some extent, such language is unavoidable, but it would be nice to have an account that is as elementary as possible. Perhaps some of the building blocks have already been implemented in (say) Sage, and it is not too hard to explain what is needed to put them together.

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I strongly disagree with the assertion that "the language of modern algebraic geometry [...] is unfamiliar to many people who might otherwise have the right skills to write the software". You are denying the existence of a flourishing research field, computational arithmetic geometry! See e.g. this paper:

Bright, M. J.; Bruin, N.; Flynn, E. V.; Logan, A., The Brauer-Manin obstruction and $\text{Ш}[2]$, LMS J. Comput. Math. 10, 354-377 (2007). ZBL1222.11084.

From the abstract: "We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA."

This isn't exactly the same question as you asked, since the relevant surfaces are intersections of two quadrics in $\mathbf{P}^4$, so not defined by a single equation; but it should serve to demonstrate that there is a substantial literature focussing on explicit algorithmic computations of the Brauer--Manin obstruction -- maybe you can find something matching your question more precisely among papers citing (or cited by) this one.

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    $\begingroup$ I think statistically the statement is correct - there exist many good programmers who don't know schemes ;) $\endgroup$
    – Achim Krause
    Sep 6, 2021 at 15:21
  • $\begingroup$ @AchimKrause Yes, you have correctly interpreted my intent. After all, I mentioned Sage, and William Stein is certainly familiar with modern algebraic geometry. To reject my assertion is to say that all those people who don't know modern algebraic geometry but consider themselves skillful programmers are not, in fact, skillful programmers. $\endgroup$
    – Timothy Chow
    Sep 6, 2021 at 16:39
  • $\begingroup$ My point is that "the right skills" are not just general-purpose programming ability. In fact many mathematicians who work in computational fields (myself included) have barely any training in software engineering per se. Implementing advanced mathematics algorithmically is a separate skill in its own right, and IMHO a severely undervalued one. $\endgroup$
    – David Loeffler
    Sep 7, 2021 at 10:16
  • $\begingroup$ @DavidLoeffler I should probably explain that I first learned about Bogdan Grechuk's project via Gasarch's blog, where Grechuk, in a kind of Polymathic spirit, invited Gasarch's readers to join him in his quest. I had those readers in mind, and they, for the most part, do not have background in algebraic geometry. I'm certainly aware of the field of computational algebraic/arithmetic geometry since that's the bread and butter of some of my colleagues whom I see every day. $\endgroup$
    – Timothy Chow
    Sep 7, 2021 at 12:54
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    $\begingroup$ I'm accepting this answer since I'm guessing it's the best answer I'm likely to get. $\endgroup$
    – Timothy Chow
    Sep 8, 2021 at 22:18

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