On octonary cubic forms

2014/02/24 by Christopher Hooley, C. Hooley · 17 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Discriminant #Field (mathematics) #Mathematical analysis #Mathematics #Pure mathematics #Riemann hypothesis #Zero (linguistics)

paper · doi:10.1112/plms/pdt066

published in Proceedings of the London Mathematical Society 109(1), 241-281 (Wiley)

openalex publication_date 2014/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

Let f ( x 1 , … , x 8 ) be a cubic form in eight variables with rational integral coefficients and non-zero discriminant. Then, assuming a Riemann hypothesis for certain Hasse–Weil L-functions, we prove that the indeterminate equation f ( x 1 , … , x 8 ) = 0 has a non-zero solution provided that the form satisfy the necessary condition that it have a non-trivial zero in every p-adic field Q p . This extends the earlier unconditional results due to Heath–Brown and the author for cubic forms in ten and nine variables.

Citations

• On the number of points on a complete intersection over a finite field
• On Waring’s problem

Cited by

• Sums of three cubes over a function field
• Rational points on cubic hypersurfaces over \mathbbF_q(t)

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