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Criterion set for diagonal forms of higher degree

  • Om Prakash
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Abstract

The 290-Theorem of Bhargava-Hanke completely classifies all universal quadratic forms over the rational integers in terms of a finite criterion set: a positive definite quadratic form is universal if and only if it represents every integer in this finite set. In this article, we study diagonal forms of higher degree over a totally real number field and prove the existence of a unique finite criterion set, minimal with respect to inclusion, for universal diagonal forms of higher degree. As part of the proof, we establish an analog of the asymptotic local-global principle for such forms and develop a local theory for the representation of integers by diagonal forms of higher degree.

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Cite

BibTeXRIS
BibTeX
@misc{prakash2026criterion,
  title = {{Criterion set for diagonal forms of higher degree}},
  author = {Om Prakash},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37736},
  eprint = {2609.37736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37736},
}
RIS
TY  - UNPB
TI  - Criterion set for diagonal forms of higher degree
AU  - Om Prakash
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37736
UR  - https://arxiv.org/abs/2609.37736
AB  - The 290-Theorem of Bhargava-Hanke completely classifies all universal quadratic forms over the rational integers in terms of a finite criterion set: a positive definite quadratic form is universal if and only if it represents every integer in this finite set. In this article, we study diagonal forms of higher degree over a totally real number field and prove the existence of a unique finite criterion set, minimal with respect to inclusion, for universal diagonal forms of higher degree. As part of the proof, we establish an analog of the asymptotic local-global principle for such forms and develop a local theory for the representation of integers by diagonal forms of higher degree.
ER  -