Systems of quadratic diophantine inequalities

Müller, Wolfgang

doi:10.5802/jtnb.488

Müller, Wolfgang ¹

¹ Institut für Statistik Technische Universität Graz 8010 Graz, Austria

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

Résumé
Abstract

Soient $Q_{1}, ..., Q_{r}$ des formes quadratiques avec des coefficients réels. Nous prouvons que pour chaque $ε > 0$ le système $| Q_{1} (x) | < ε, ..., | Q_{r} (x) | < ε$ des inégalités a une solution entière non-triviale si le système $Q_{1} (x) = 0, ..., Q_{r} (x) = 0$ a une solution réelle non-singulière et toutes les formes $\sum_{i = 1}^{r} α_{i} Q_{i}$ , $α = (α_{1}, ..., α_{r}) \in ℝ^{s}, α \neq 0$ sont irrationnelles avec rang $> 8 r$ .

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

MR Zbl

DOI : 10.5802/jtnb.488

Affiliations des auteurs :

Müller, Wolfgang ¹

¹ Institut für Statistik Technische Universität Graz 8010 Graz, Austria

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     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},
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     year = {2005},
     doi = {10.5802/jtnb.488},
     zbl = {1082.11020},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.488/}
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Müller, Wolfgang. 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. http://www.numdam.org/articles/10.5802/jtnb.488/

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