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The asymmetric product of three inhomogeneous linear forms

Published online by Cambridge University Press:  09 April 2009

V. K. Grover
V. K. Grover
Affiliation:
Centre for Advanced Study in Mathematics, Panjab UniversityChandigarh-160 014, India
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Abstract

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Let Λ be a lattice in R3 of determinant 1. Define the homogeneous minium of Λ as mn (Λ) = inf |u1, u2, u3| extended over all points (u1, u2, u3) of Λ other than the origin. It is shown that for any given (c1, c2, c3) in R3 there exists a point (u1, u2, u3) of Λ for which provided that ρσ > 1/64 if mn (Λ) = 0, and ρσ ≥1/16.81 if mn (A) > 0.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 236 - 250
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bambah, R. P. and Woods, A. C., ‘Minkowski's conjecture of n = 5, A theorem of Skubenko’, J. Number Theory 12 (1980), 27–48.CrossRefGoogle Scholar
[2]Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘On the inhomogeneous minimum of the product of n-linear forms’, Mathematika 3 (1956), 25–39.CrossRefGoogle Scholar
[3]Chalk, J. H. H., ‘On the positive values of linear forms’, Quart. J. Math. Oxford Ser. 18 (1947), 215–227.CrossRefGoogle Scholar
[4]Davenport, H., ‘Note on the product of three homogeneous linear forms’, J. London Math. Soc. 14 (1941), 98–101.CrossRefGoogle Scholar
[5]Davenport, H., ‘Non-homogeneous ternary quadratic forms’, Acta Math. 80 (1948), 65–95.CrossRefGoogle Scholar
[6]Grover, V. K., ‘Asymmetric inequalities for non-homogeneous forms’, Ph. D. thesis, 1979.Google Scholar
[7]Mahler, K., ‘On lattice points in n-dimensional star bodies I, Existence theorems’, Proc. Roy. Soc. London Ser A 187 (1946), 151–187.Google Scholar
[8]Remak, R., ‘Verallgemeinerung eines Minkowskischen Satzes I, II’, Math. Z. 17 (1923), 1–34, 18 (1923), 173–200.CrossRefGoogle Scholar
[9]Woods, A. C., ‘The asymmetric product of three inhomogeneous linear forms’, J. Austral. Math. Soc. Ser. A 31 (1981), 439–455.CrossRefGoogle Scholar