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Systems of cubic forms in many variables

  • Simon L. Rydin Myerson ORCID logo EMAIL logo

Abstract

We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided n25R, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in 25R variables, previous work having required that nR2. One conjectures that n6R+1 should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

Award Identifier / Grant number: EP/J500495/1

Award Identifier / Grant number: EP/M507970/1

Funding statement: This research was supported by Engineering and Physical Sciences Research Council grants EP/J500495/1 and EP/M507970/1.

Acknowledgements

This paper is based on a DPhil thesis submitted to Oxford University. I would like to thank my DPhil supervisor, Roger Heath-Brown. I am grateful to Rainer Dietmann for helpful conversations.

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Received: 2017-02-27
Revised: 2017-08-29
Published Online: 2017-10-17
Published in Print: 2019-12-01

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