[#P27] Hilbert's tenth problem over the rationals
Problem. Does there exist an algorithm which, given any polynomial \(f\in\mathbb{Z}[x_1,\ldots,x_n]\), decides whether the equation \(f(x_1,\ldots,x_n)=0\) has a solution in \(\mathbb{Q}^n\)?
1Context
The corresponding problem over the integers is undecidable. Replacing integers with rational numbers changes the problem enough that its decidability remains unknown.
2Problem setup
Definition 1 (A decision algorithm must halt on every input and answer correctly whether a rational solution exists). A decision algorithm must halt on every input and answer correctly whether a rational solution exists.
Definition 2 (A multivariable polynomial equation has a rational solution when rational values for all variables make the polynomial equal to zero). A multivariable polynomial equation has a rational solution when rational values for all variables make the polynomial equal to zero.
Remark 1. The corresponding problem over the integers is undecidable. Replacing integers with rational numbers changes the problem enough that its decidability remains unknown.
3What counts as a solution
- Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Koenigsmann gives a universal first-order definition of Z in Q and proves that the universal-existential theory of Q is undecidable. The purely existential decision problem asked by Hilbert's tenth problem over Q remains open. Exact unresolved remainder: Give a terminating algorithm deciding rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.[2][1]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
Original intake status. The cited survey identifies Hilbert's tenth problem over Q as unresolved. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Undecidability over the integers does not decide the rational case. Many subrings and conditional cases are known.
Recorded example 1. The equation x^2 + y^2 = 1 has rational solutions, including x = 3/5 and y = 4/5.
Computational notes
- Searching rational points by bounded height can find solutions but cannot certify their absence in general.
2See also
How to cite
TheoremDB contributors, “Hilbert's tenth problem over the rationals,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/hilberts-tenth-problem-over-the-rationalsThis page as plain text: hilberts-tenth-problem-over-the-rationals.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Sylvy Anscombe, Valentijn Karemaker, Zeynep Kisakürek, Vlerë Mehmeti, Margherita Pagano, and Laura Paladino, “A survey of local-global methods for Hilbert's Tenth Problem”. arXiv:2309.14987 (2023). Sylvy Anscombe et al., arXiv:2309.14987, abstract and introduction. ↗preprint · primary source · arXiv:2309.14987, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited survey identifies Hilbert's tenth problem over Q as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and introduction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the exact Q-decidability problem as open and summarizes current local-global methods.Source named by the research packet.
- Jochen Koenigsmann, “Defining \mathbb Z in \mathbb Q”. Annals of Mathematics (2016), 73-93. DOI 10.4007/annals.2016.183.1.2. Theorem 1 and Corollary 3 on pages 74-75. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Universally defines Z inside Q and proves universal-existential undecidability without settling the purely existential theory.
An original CC0 restatement prepared by TheoremDB maintainers.