Bihomogeneous forms in many variables

We count integer points on bihomogeneous varieties using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

[1]  Craig V. Spencer.  The Manin conjecture for x0y0+⋯+xsys=0☆ , 2009 .

[2]  M. Robbiani.  On the Number of Rational Points of Bounded Height on Smooth Bilinear Hypersurfaces in Biprojective Space , 2001 .

[3]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[4]  W. Schmidt.  The density of integer points on homogeneous varieties , 1985 .

[5]  R. Vaughan.  The Hardy-Littlewood Method , 1981 .

[6]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[7]  B. Birch.  Forms in many variables , 1962, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[8]  H. Davenport.  Cubic forms in thirty-two variables , 1959, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  H. Davenport,et al.  Analytic Methods for Diophantine Equations and Diophantine Inequalities: Waring's problem: history , 2005 .