Solubility of Additive Forms of Twice Odd Degree over ℚ 2 (5)
Drew Duncan; David B. Leep1
1Department of Mathematics University of Kentucky 719 Patterson Office Tower Lexington, Kentucky 40506-0027 USA
Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 1, pp. 293-309

We prove that an additive form of degree d=2m, m odd, m≥3, over the unramified quadratic extension ℚ 2 (5) has a nontrivial zero if the number of variables s satisifies s≥4d+1. If 3∤d, then there exists a nontrivial zero if s≥3 2d+1, this bound being optimal. We give examples of forms in 3d variables without a nontrivial zero in case that 3∣d.

Nous prouvons qu’une forme additive de degré d=2m, m impair, m≥3, sur l’extension quadratique non ramifiée ℚ 2 (5) admet un zéro non trivial si le nombre de variables s satisfait la condition s≥4d+1. Si 3∤d, alors il existe un zéro non trivial si s≥3 2d+1, cette borne étant optimale. Si 3∣d, nous donnons des exemples de formes en 3d variables n’ayant pas de zéros non triviaux.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1279
Classification: 11D72, 11D88, 11E76
Keywords: Forms in many variables, p-adic fields, unramified extension, additive forms

Drew Duncan  ; David B. Leep  1

1 Department of Mathematics University of Kentucky 719 Patterson Office Tower Lexington, Kentucky 40506-0027 USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Drew Duncan; David B. Leep. Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$. Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 1, pp. 293-309. doi: 10.5802/jtnb.1279

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