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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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.
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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