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arXiv · 2609.37736

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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Om Prakash. 2026-09-29. Criterion set for diagonal forms of higher degree. https://arxiv.org/abs/2609.37736

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