Systems of diagonal Diophantine inequalities
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- by Eric Freeman PDF
- Trans. Amer. Math. Soc. 355 (2003), 2675-2713 Request permission
Abstract:
We treat systems of real diagonal forms $F_1(\mathbf {x}), F_2(\mathbf {x}), \ldots , F_R(\mathbf {x})$ of degree $k$, in $s$ variables. We give a lower bound $s_0(R,k)$, which depends only on $R$ and $k$, such that if $s \geq s_0(R,k)$ holds, then, under certain conditions on the forms, and for any positive real number $\epsilon$, there is a nonzero integral simultaneous solution $\mathbf {x} \in \mathbb {Z}^s$ of the system of Diophantine inequalities $|F_i(\mathbf {x})| < \epsilon$ for $1 \leq i \leq R$. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.References
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Additional Information
- Eric Freeman
- Affiliation: Department of Mathematics, University of Colorado, 395 UCB, Boulder, Colorado 80309
- Address at time of publication: School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
- Email: freem@ias.edu
- Received by editor(s): October 15, 2001
- Published electronically: March 17, 2003
- Additional Notes: The author was supported by an NSF Postdoctoral Fellowship.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2675-2713
- MSC (2000): Primary 11D75; Secondary 11D41, 11D72, 11P55
- DOI: https://doi.org/10.1090/S0002-9947-03-03121-0
- MathSciNet review: 1975395