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 of degree , in variables. We give a lower bound , which depends only on and , such that if holds, then, under certain conditions on the forms, and for any positive real number , there is a nonzero integral simultaneous solution of the system of Diophantine inequalities for . 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