Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 217-236.

Soient ${Q}_{1},...,{Q}_{r}$ des formes quadratiques avec des coefficients réels. Nous prouvons que pour chaque $\epsilon >0$ le système $|{Q}_{1}\left(x\right)|<\epsilon ,...,|{Q}_{r}\left(x\right)|<\epsilon$ des inégalités a une solution entière non-triviale si le système ${Q}_{1}\left(x\right)=0,...,{Q}_{r}\left(x\right)=0$ a une solution réelle non-singulière et toutes les formes ${\sum }_{i=1}^{r}{\alpha }_{i}{Q}_{i}$, $\alpha =\left({\alpha }_{1},...,{\alpha }_{r}\right)\in {ℝ}^{s},\alpha \ne 0$ sont irrationnelles avec rang $>8r$.

Let ${Q}_{1},\cdots ,{Q}_{r}$ be quadratic forms with real coefficients. We prove that for any $ϵ>0$ the system of inequalities $|{Q}_{1}\left(x\right)|<ϵ,\cdots ,|{Q}_{r}\left(x\right)|<ϵ$ has a nonzero integer solution, provided that the system ${Q}_{1}\left(x\right)=0,\cdots ,{Q}_{r}\left(x\right)=0$ has a nonsingular real solution and all forms in the real pencil generated by ${Q}_{1},\cdots ,{Q}_{r}$ are irrational and have rank $>8r$.

Publié le :
DOI : https://doi.org/10.5802/jtnb.488
@article{JTNB_2005__17_1_217_0,
author = {Wolfgang M\"uller},
title = {Systems of quadratic diophantine inequalities},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {217--236},
publisher = {Universit\'e Bordeaux 1},
volume = {17},
number = {1},
year = {2005},
doi = {10.5802/jtnb.488},
zbl = {1082.11020},
mrnumber = {2152222},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/}
}
Wolfgang Müller. Systems of quadratic diophantine inequalities. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 217-236. doi : 10.5802/jtnb.488. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.488/

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