Schindler, Damaris Bihomogeneous forms in many variables. (English. French summary) Zbl 1425.11055 J. Théor. Nombres Bordx. 26, No. 2, 483-506 (2014). Summary: 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. Cited in 1 ReviewCited in 8 Documents MSC: 11D45 Counting solutions of Diophantine equations 11D72 Diophantine equations in many variables 11P55 Applications of the Hardy-Littlewood method Keywords:Hardy-Littlewood method; bihomogeneous equations; counting functions PDFBibTeX XMLCite \textit{D. Schindler}, J. Théor. Nombres Bordx. 26, No. 2, 483--506 (2014; Zbl 1425.11055) Full Text: DOI arXiv References: [1] B. J. Birch, Forms in many variables. Proc. Roy. Soc. Ser. A 265, (1961), 245-263. · Zbl 0103.03102 [2] H. Davenport, Cubic Forms in Thirty-Two Variables. Phil. Trans. R. Soc. Lond. A 251, (1959), 193-232. · Zbl 0084.27202 [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. · Zbl 1125.11018 [4] J. Harris, Algebraic Geometry, A First Course. Springer, (1993). · Zbl 0779.14001 [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. · Zbl 1020.11046 [6] W. M. Schmidt, Simultaneous rational zeros of quadratic forms. Seminar Delange-Pisot-Poitou 1981. Progress in Math. 22, (1982), 281-307. · Zbl 0492.10017 [7] W. M. Schmidt, The density of integer points on homogeneous varieties. Acta Math. 154, 3-4, (1985), 243-296. · Zbl 0561.10010 [8] C. V. Spencer, The Manin conjecture for \(x_0y_0+\ldots +x_sy_s=0\). J. Number Theory 129, 6, (2009), 1505-1521. · Zbl 1171.11054 [9] K. van Valckenborgh, Squareful numbers in hyperplanes. arXiv 1001.3296v3. · Zbl 1321.11038 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.