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Systems of diagonal Diophantine inequalities


Author: Eric Freeman
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
Published electronically: March 17, 2003
MathSciNet review: 1975395
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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 $\displaystyle{{\mathbf x}\in {\mathbb Z}^s}$ of the system of Diophantine inequalities $\vert F_i({\mathbf x})\vert < \epsilon$ 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.


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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

DOI: https://doi.org/10.1090/S0002-9947-03-03121-0
Keywords: Combined systems of Diophantine equations and inequalities, forms in many variables, applications of the Hardy-Littlewood method.
Received by editor(s): October 15, 2001
Published electronically: March 17, 2003
Additional Notes: The author was supported by an NSF Postdoctoral Fellowship.
Article copyright: © Copyright 2003 American Mathematical Society

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