Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 217-236.

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 $>8r$.

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 $>8r$.

DOI: 10.5802/jtnb.488
Müller, Wolfgang 1

1 Institut für Statistik Technische Universität Graz 8010 Graz, Austria
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Müller, Wolfgang. Systems of quadratic diophantine inequalities. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 217-236. doi : 10.5802/jtnb.488. http://www.numdam.org/articles/10.5802/jtnb.488/

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