# P27: Hilbert's tenth problem over the rationals

- ID: `P27`
- Reference: `hilberts-tenth-problem-over-the-rationals`
- Page: https://theoremdb.org/statements/P27
- Record maturity: Reviewed problem with recorded work

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

### Context

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

### Problem setup

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

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: The cited survey identifies Hilbert's tenth problem over Q as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited 2023 survey on 2026-07-22.
- Undecidability over the integers does not decide the rational case. Many subrings and conditional cases are known.

- Recorded example: The equation x^2 + y^2 = 1 has rational solutions, including x = 3/5 and y = 4/5.

### Open directions

- **Route 1** (reported): Give and prove correct a terminating algorithm for rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists. [1](#reference-1)

### Computational notes

- Searching rational points by bounded height can find solutions but cannot certify their absence in general.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hilberts-tenth-problem-over-the-rationals`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2309.14987
   - Also cited at abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2309.14987, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.4007/annals.2016.183.1.2
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Universally defines Z inside Q and proves universal-existential undecidability without settling the purely existential theory.