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Ten unknowns for Hilbert's tenth problem over the integers
Abstract
Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation P(z1,…,z10)=0 (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.
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BibTeX
@misc{sun2026unknowns,
title = {{Ten unknowns for Hilbert's tenth problem over the integers}},
author = {Zhi-Wei Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01594},
eprint = {2609.01594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01594},
}
RIS
TY - UNPB
TI - Ten unknowns for Hilbert's tenth problem over the integers
AU - Zhi-Wei Sun
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01594
UR - https://arxiv.org/abs/2609.01594
AB - Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation P(z_1,…,z_10)=0 (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.
ER -