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

Duncan, Drew; Leep, David B.

doi:10.5802/jtnb.1279

Solubility of Additive Forms of Twice Odd Degree over

ℚ_{2} (\sqrt{5})

Duncan, Drew ; Leep, David B.¹

¹Department 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

Abstract (VO)
Résumé

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 $3 ∣ d$ .

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.

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

MR Zbl

DOI : 10.5802/jtnb.1279

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

Affiliations des auteurs :

Duncan, Drew ; Leep, David B. 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

@article{JTNB_2024__36_1_293_0,
     author = {Duncan, Drew and Leep, David B.},
     title = {Solubility of {Additive} {Forms} of {Twice} {Odd} {Degree} over $\mathbb{Q}_2(\sqrt{5})$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {293--309},
     year = {2024},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {1},
     doi = {10.5802/jtnb.1279},
     mrnumber = {4788374},
     zbl = {07892785},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.1279/}
}

TY  - JOUR
AU  - Duncan, Drew
AU  - Leep, David B.
TI  - Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2024
SP  - 293
EP  - 309
VL  - 36
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://www.numdam.org/articles/10.5802/jtnb.1279/
DO  - 10.5802/jtnb.1279
LA  - en
ID  - JTNB_2024__36_1_293_0
ER  -

%0 Journal Article
%A Duncan, Drew
%A Leep, David B.
%T Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$
%J Journal de théorie des nombres de Bordeaux
%D 2024
%P 293-309
%V 36
%N 1
%I Société Arithmétique de Bordeaux
%U https://www.numdam.org/articles/10.5802/jtnb.1279/
%R 10.5802/jtnb.1279
%G en
%F JTNB_2024__36_1_293_0

Duncan, Drew; Leep, David B. 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

Bibliographie
Cité par

[1] Davenport, Harold; Lewis, Donald J. Homogeneous additive equations, Proc. R. Soc. Lond., Ser. A, Volume 274 (1963), pp. 443-460 | DOI | MR | Zbl

[2] Duncan, Drew; Leep, David B. Solubility of Additive Forms of Twice Odd Degree over Ramified Quadratic Extensions of $ℚ_{2}$ , Acta Arith., Volume 201 (2021) no. 2, pp. 149-164 | DOI | MR

[3] Duncan, Drew; Leep, David B. 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 | MR

[4] Leep, David B.; Sordo Vieira, Luis Diagonal equations over unramified extensions of $ℚ_{p}$ , Bull. Lond. Math. Soc., Volume 50 (2018) no. 4, pp. 619-634 | DOI | MR | Zbl

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

[6] de Paula Miranda, Bruno; Godinho, Hemar; Knapp, Michael P. Diagonal forms over quadratic extensions of $ℚ_{2}$ , Publ. Math. Debr., Volume 101 (2022) no. 1-2, pp. 63-101 | MR | DOI | Zbl

Cité par Sources :