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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inequalities for decomposable forms of degree $n+1$ in $n$ variables


Author: Jeffrey Lin Thunder
Journal: Trans. Amer. Math. Soc. 354 (2002), 3855-3868
MSC (2000): Primary :, 11D75, 11D45; Secondary :, 11D72
Published electronically: June 10, 2002
MathSciNet review: 1926855
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Abstract: We consider the number of integral solutions to the inequality $\vert F(\mathbf{x}) \vert\le m$, where $F(\mathbf{X} )\in \mathbb{Z} [\mathbf{X} ]$ is a decomposable form of degree $n+1$ in $n$ variables. We show that the number of such solutions is finite for all $m$ only if the discriminant of $F$ is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.


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Additional Information

Jeffrey Lin Thunder
Affiliation: Department of Mathematics, Northern Illinois University, DeKalb, Illinois 60115
Email: jthunder@math.niu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03038-6
PII: S 0002-9947(02)03038-6
Received by editor(s): October 24, 2000
Published electronically: June 10, 2002
Additional Notes: Research partially supported by NSF grant DMS-9800859
Article copyright: © Copyright 2002 American Mathematical Society



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