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Journal of the London Mathematical Society Advance Access published online on March 10, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm123
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© 2008 London Mathematical Society

Systems of cubic forms

Rainer Dietmann

Institut für Algebra und Zahlentheorie
Pfaffenwaldring 57
D-70569 Stuttgart
Germany

We discuss the existence of rational and p-adic zeros of systems of cubic forms. In particular, we prove that for p!=2 any system of r cubic forms over Qp in more than 125r3+705r2+210r variables admits a non-trivial p-adic zero, and that any system of r rational cubic forms in more than O(r4 m6+r6 m5) variables admits a rational linear space of zeros of dimension at least m.

2000 Mathematics Subject Classification 11D25, 11D72, 11D88, 11E76.

Received September 21, 2006; revised October 10, 2007;
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