[#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 \(\mathbb{Q}^n\)?

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1Context

The corresponding problem over the integers is undecidable. Replacing integers with rational numbers changes the problem enough that its decidability remains unknown.

2Remarks

Remark 1. A decision algorithm must halt on every input and answer correctly whether a rational solution exists.

Remark 2. A multivariable polynomial equation has a rational solution when rational values for all variables make the polynomial equal to zero.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Hilbert's tenth problem over the rationals, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.^[1] Relevance to this problem For Hilbert's tenth problem over the rationals, record hilberts-tenth-problem-over-the-rationals-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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.

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Plain text

“Hilbert's tenth problem over the rationals.” TheoremDB. P27. Problem statement; statement text SHA-256 1f57419ef7e3f9fd2ff1bb4edd2a64a148e558c6dc5626746a3c270f9fadb7cf. https://theoremdb.org/statement/?ref=P27

BibTeX

@misc{theoremdb-problem-1f57419ef7e3f9fd2ff1bb4edd2a64a148e558c6dc5626746a3c270f9fadb7cf,
  title = {{Hilbert's tenth problem over the rationals}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 1f57419ef7e3f9fd2ff1bb4edd2a64a148e558c6dc5626746a3c270f9fadb7cf},
  url = {https://theoremdb.org/statement/?ref=P27}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P27] Hilbert's tenth problem over the rationals

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