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Hilbert's tenth problem over the rational numbers

OpenAI · September 24, 2026

Earlier edition

This is an earlier edition (September 24, 2026). Current edition: Hilbert’s tenth problem over the rational numbers, October 6, 2026.

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@misc{OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026,
  author = {{OpenAI}},
  title = {{Hilbert's tenth problem over the rational numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf}{OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026}},
  year = {2026}
}

Family 004: Hilbert’s tenth problem over ℚ

Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.

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