Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$

Drew Duncan; David B. Leep

doi:10.5802/jtnb.1279

Solubility of Additive Forms of Twice Odd Degree over

ℚ_{2} (\sqrt{5})

Drew Duncan ; David B. Leep ¹

¹ Department of Mathematics University of Kentucky 719 Patterson Office Tower Lexington, Kentucky 40506-0027 USA

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 293-309.

Résumé
Abstract

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

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

Reçu le : 2022-10-14
Révisé le : 2023-05-17
Accepté le : 2023-06-19
Publié le : 2024-07-25

DOI : 10.5802/jtnb.1279

Classification : 11D72, 11D88, 11E76
Mots clés : Forms in many variables, p-adic fields, unramified extension, additive forms

Affiliations des auteurs :

Drew Duncan ; David B. Leep ¹

¹ Department of Mathematics University of Kentucky 719 Patterson Office Tower Lexington, Kentucky 40506-0027 USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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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, Tome 36 (2024) no. 1, pp. 293-309. doi : 10.5802/jtnb.1279. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1279/

Bibliographie
Cité par

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[3] Drew Duncan; David B. Leep Solubility of Additive Quartic Forms over Ramified Quadratic Extensions of $ℚ_{2}$ , Int. J. Number Theory, Volume 18 (2022) no. 10, pp. 2265-2278 | DOI | Zbl

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[5] Bruno de Paula Miranda Diagonal forms over the unramified quadratic extension of $ℚ_{2}$ , Ph. D. Thesis, Universidade de Brasília (2022)

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