Hilbert's tenth problem over the rational numbers
Hilbert's tenth problem over the rational numbers asks whether there is an algorithm that decides, given a polynomial with integer coefficients (the number of variables being part of the input), whether it has a zero with rational coordinates. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a negative answer. This page summarizes the published claim and lists publicly available follow-up work.
Background
- According to the manuscript's introduction, Hilbert's tenth problem over the integers was answered negatively by the work of Davis, Putnam and Robinson (1961) and Matiyasevich (1970): every recursively enumerable relation on the positive integers is Diophantine. Over the rationals, a Diophantine definition of the integers, or a Diophantine model of integer arithmetic in a power of the rationals, would transfer this result.
- The same introduction recalls that J. Robinson (1949) defined the integers in the rationals by a first-order formula; Poonen (Characterizing integers among rational numbers with a universal-existential formula) reduced it to a ∀∃ formula; and Koenigsmann (Defining Z in Q, Annals of Mathematics) gave a universal definition. These do not supply the existential definition the transfer needs. Mazur's conjecture on the topology of rational points, as observed by Mazur and by Cornelissen and Zahidi, would rule out a Diophantine definition of the integers and a Diophantine model of integer arithmetic.
- For rings of integers of number fields, the introduction cites the recent unconditional results of Koymans and Pagano (Hilbert's tenth problem via additive combinatorics) and of Alpöge, Bhargava, Ho and Shnidman (Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field), and notes that the field of rational numbers is not covered by them.
The original result
- Claim. There is no algorithm which, given a polynomial f ∈ Z[X1, …, Xn], with n part of the input, decides whether f has a zero in Qn (Theorem 1.1).
- Source. Hilbert's tenth problem over the rational numbers in the openai/math repository (commit
1581d4c). - Formalization. The repository's contents list gives no Lean scope note for this family, and no Comparator statement is listed for it.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 16 October 2026) |
|---|---|---|---|---|
| 2026-10-14 (submitted) | Z.-W. Sun, 86 unknowns for Hilbert's tenth problem over the rationals (arXiv:2610.20691) | Extension (number of unknowns) | The abstract states that Hilbert's tenth problem over the rationals "was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction", and builds on that solution to bound the number of unknowns needed for undecidability; the title gives 86. | arXiv preprint (42 pages). |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.