Bihomogeneous forms in many variables

Damaris Schindler[1]

  • [1] Hausdorff Center for Mathematics Endenicher Allee 62 53115 Bonn, Germany

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 2, page 483-506
  • ISSN: 1246-7405

Abstract

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We count integer points on varieties given by bihomogeneous equations 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.

How to cite

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Schindler, Damaris. "Bihomogeneous forms in many variables." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 483-506. <http://eudml.org/doc/275801>.

@article{Schindler2014,
abstract = {We count integer points on varieties given by bihomogeneous equations 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.},
affiliation = {Hausdorff Center for Mathematics Endenicher Allee 62 53115 Bonn, Germany},
author = {Schindler, Damaris},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {10},
number = {2},
pages = {483-506},
publisher = {Société Arithmétique de Bordeaux},
title = {Bihomogeneous forms in many variables},
url = {http://eudml.org/doc/275801},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Schindler, Damaris
TI - Bihomogeneous forms in many variables
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/10//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 2
SP - 483
EP - 506
AB - We count integer points on varieties given by bihomogeneous equations 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.
LA - eng
UR - http://eudml.org/doc/275801
ER -

References

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  1. B. J. Birch, Forms in many variables. Proc. Roy. Soc. Ser. A 265, (1961), 245–263. Zbl0103.03102MR150129
  2. H. Davenport, Cubic Forms in Thirty-Two Variables. Phil. Trans. R. Soc. Lond. A 251, (1959), 193–232. Zbl0084.27202MR105394
  3. H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, (2005). With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, Edited and prepared for publication by T. D. Browning. Zbl1125.11018MR2152164
  4. J. Harris, Algebraic Geometry, A First Course. Springer, (1993). Zbl0779.14001MR1416564
  5. M. Robbiani, On the number of rational points of bounded height on smooth bilinear hypersurfaces in biprojective space. J. London Math. Soc. 63, (2001), 33–51. Zbl1020.11046MR1801715
  6. W. M. Schmidt, Simultaneous rational zeros of quadratic forms. Seminar Delange-Pisot-Poitou 1981. Progress in Math. 22, (1982), 281–307. Zbl0492.10017MR693325
  7. W. M. Schmidt, The density of integer points on homogeneous varieties. Acta Math. 154, 3-4, (1985), 243–296. Zbl0561.10010MR781588
  8. C. V. Spencer, The Manin conjecture for x 0 y 0 + ... + x s y s = 0 . J. Number Theory 129, 6, (2009), 1505–1521. Zbl1171.11054MR2521490
  9. K. van Valckenborgh, Squareful numbers in hyperplanes. arXiv 1001.3296v3. Zbl1321.11038MR2968632

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