Editing Hilbert's tenth problem (section)

==Extensions of Hilbert's tenth problem==
[[File:CornelissenGunther ShlapentokhAlexandra MatiyasevichYuri MFO6700.jpg|thumb|160px|Alexandra Shlapentokh (middle) in 2003]]
Although Hilbert posed the problem for the rational integers, it can be just as well asked for many [[ring (mathematics)|ring]]s (in particular, for any ring whose number of elements is [[countable]]). Obvious examples are the rings of integers of [[algebraic number field]]s as well as the [[rational number]]s.

There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields. Basing themselves on earlier work by [[Jan Denef]] and Leonard Lipschitz and using class field theory, Harold N. Shapiro and [[Alexandra Shlapentokh]] were able to prove:
<blockquote>''Hilbert's tenth problem is unsolvable for the ring of integers of any algebraic number field whose [[Galois group]] over the rationals is [[abelian group|abelian]].''</blockquote>

Shlapentokh and Thanases Pheidas (independently of one another) obtained the same result for algebraic number fields admitting exactly one pair of complex conjugate embeddings.

The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open. Likewise, despite much interest, the problem for equations over the rationals remains open. [[Barry Mazur]] has conjectured that for any [[Algebraic variety|variety]] over the rationals, the topological closure over the reals of the set of solutions has only finitely many components.<ref>{{cite journal
 | last = Poonen | first = Bjorn | author-link = Bjorn Poonen
 | doi = 10.1090/S0894-0347-03-00433-8
 | issue = 4
 | journal = [[Journal of the American Mathematical Society]]
 | mr = 1992832
 | pages = 981–990
 | title = Hilbert's tenth problem and Mazur's conjecture for large subrings of <math>\mathbb{Q}</math>
 | url = https://klein.mit.edu/~poonen/papers/subrings.pdf
 | volume = 16
 | year = 2003| s2cid = 8486815 }}</ref> This conjecture implies that the integers are not Diophantine over the rationals, and so if this conjecture is true, a negative answer to Hilbert's Tenth Problem would require a different approach than that used for other rings.

In 2024, Peter Koymans and Carlo Pagano published a claimed proof that Hilbert’s 10th problem is undecidable for every ring of integers using [[additive combinatorics]].<ref name=wired-rings>{{Cite magazine |last=Howlett |first=Joseph |title=New Proofs Expand the Limits of What Cannot Be Known |url=https://www.wired.com/story/new-proofs-expand-the-limits-of-what-cannot-be-known/?utm_source=firefox-newtab-en-gb |access-date=2025-03-14 |magazine=Wired |language=en-US |issn=1059-1028}}</ref><ref>{{Citation |last1=Koymans |first1=Peter |title=Hilbert's tenth problem via additive combinatorics |date=2024-12-02 |url=https://arxiv.org/abs/2412.01768 |access-date=2025-03-14 |arxiv=2412.01768 |last2=Pagano |first2=Carlo}}</ref> Another team of mathematicians subsequently claimed another proof of the same result, using different methods.<ref name=wired-rings/><ref>{{Citation |last1=Alpöge |first1=Levent |title=Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field |date=2025-01-30 |url=https://arxiv.org/abs/2501.18774 |access-date=2025-03-14 |arxiv=2501.18774 |last2=Bhargava |first2=Manjul |last3=Ho |first3=Wei |last4=Shnidman |first4=Ari}}</ref>