Abstract
We give a variant of Weyl’s inequality for systems of forms together with applications. First we use this to give a different formulation of a theorem of B.J. Birch on forms in many variables. More precisely, we show that the dimension of the locus V ∗ introduced in this work can be replaced by the maximal dimension of the singular loci of forms in the linear system of the given forms. In some cases this improves on the aforementioned theorem of Birch.
Second, we improve on a theorem of W.M. Schmidt which states that the number of integer points inside a given box, that lie on the variety given by a system of homogeneous forms of the same degree, satisfies the asymptotic behaviour as predicted by the classical circle method, as soon as the so called h-invariant of the system is sufficiently large. In this direction we generalise previous improvements of R. Dietmann on systems of quadratic and cubic forms to systems of forms of general degree.
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References
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Acknowledgements
The author would like to thank Prof. T.D. Browning for comments on an earlier version of this paper, Prof. P. Salberger for helpful discussions, and the referees for a careful reading of the manuscript and their suggestions.
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Schindler, D. (2015). A Variant of Weyl’s Inequality for Systems of Forms and Applications. In: Alaca, A., Alaca, Ş., Williams, K. (eds) Advances in the Theory of Numbers. Fields Institute Communications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3201-6_9
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DOI: https://doi.org/10.1007/978-1-4939-3201-6_9
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