Nous prouvons qu’une forme additive de degré , impair, , sur l’extension quadratique non ramifiée admet un zéro non trivial si le nombre de variables satisfait la condition . Si , alors il existe un zéro non trivial si , cette borne étant optimale. Si , nous donnons des exemples de formes en variables n’ayant pas de zéros non triviaux.
We prove that an additive form of degree , odd, , over the unramified quadratic extension has a nontrivial zero if the number of variables satisifies . If , then there exists a nontrivial zero if , this bound being optimal. We give examples of forms in variables without a nontrivial zero in case that .
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Mots clés : Forms in many variables, p-adic fields, unramified extension, additive forms
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@article{JTNB_2024__36_1_293_0, author = {Drew Duncan and David B. Leep}, 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}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {36}, number = {1}, year = {2024}, doi = {10.5802/jtnb.1279}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1279/} }
TY - JOUR AU - Drew Duncan AU - David B. Leep 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://jtnb.centre-mersenne.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 Drew Duncan %A David B. Leep %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://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1279/ %R 10.5802/jtnb.1279 %G en %F JTNB_2024__36_1_293_0
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/
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