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Raka, Madhu
- Refined Estimates on Conjectures of Woods and Minkowski
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Authors
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1 Centre for Advanced Study in Mathematics Panjab University, Chandigarh-160014, IN
1 Centre for Advanced Study in Mathematics Panjab University, Chandigarh-160014, IN
Source
The Journal of the Indian Mathematical Society, Vol 83, No 1-2 (2016), Pagination: 61-85Abstract
Let Λ be a lattice in Rn reduced in the sense of Korkine and Zolotareff having a basis of the form (A1, 0, 0, . . . , 0), (a2,1,A2, 0, . . . , 0), . . . , (an,1, an,2, . . . , an,n−1,An) where A1,A2, . . . ,An are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if A1A2· · ·An = 1 and Ai ≤ A1 for each i then any closed sphere in Rn of radius √n/2 contains a point of Λ. Woods' Conjecture is known to be true for n ≤ 9. In this paper we give estimates on the Conjecture of Woods for 10 ≤ n ≤ 33, improving the earlier best known results of Hans-Gill et al. These lead to an improvement, for these values of n, to the estimates on the long standing classical conjecture of Minkowski on the product of n non-homogeneous linear forms.Keywords
Lattice, Covering, Non-Homogeneous, Product of Linear Forms, Critical Determinant, Korkine and Zolotareff Reduction, Hermite’s Constant, Center Density.
References
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- R. J. Hans-Gill, Madhu Raka and Ranjeet Sehmi, Estimates On Conjectures of Minkowski and Woods, Indian Jl. Pure Appl. Math.,41(4) (2010), 595-606.
- R.J. Hans-Gill, Madhu Raka and Ranjeet Sehmi, On Conjectures of Minkowski and Woods for n = 8, Acta Arithmetica, 147(4) (2011), 337-385.
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- Leetika Kathuria and Madhu Raka, On Conjectures of Minkowski and Woods for n = 9 , arXiv:1410.5743v1 [math.NT], 21 Oct, 2014, To appear in Proc. Math. Indian Academy of sciences.
- Leetika Kathuria and Madhu Raka, Refined Estimates on Conjectures of Woods and Minkowski, arXiv:1501.03277v1 [math.NT], 14 Jan 2015.
- Leetika Kathuria and Madhu Raka, Generalization of a result of Birch and SwinnertonDyer, Submitted for publication
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- A Generalization of a Result of Birch and Swinnerton-Dyer
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Authors
Affiliations
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
Source
The Journal of the Indian Mathematical Society, Vol 85, No 3-4 (2018), Pagination: 342-355Abstract
In this paper, we give a proof of the generalization of a result of Birch and Swinnerton-Dyer [1956], which has been used by Hans-Gill, Sehmi and authors while obtaining estimates on the classical conjecture of Minkowski on the product of n non-homogeneous linear forms.Keywords
Minkowski's Conjecture, Lattices, Homogeneous Minimum, Non-Homogeneous, Linear Forms, Unimodular Transformation.References
- Bambah, R. P., Dumir, V. C. and Hans-Gill, R. J., Non-homogeneous problems: Conjectures of Minkowski and Watson, Number Theory, Trends in Mathematics, Birkhauser Verlag, Basel, (2000) 15-41.
- 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.
- Cebotarev, N., Beweis des Minkowski'schen Satzes uber lineare inhomogene Formen, Vierteljschr. Naturforsch. Ges. Zurich, 85 Beiblatt, (1940), 27-30.
- Gruber, P., Convex and discrete geometry, Springer Grundlehren Series, Vol. 336, 2007.
- Hans-Gill, R. J., Madhu Raka and Ranjeet Sehmi, On conjectures of Minkowski and Woods for n = 7, J. Number Theory, 129 (2009), 1011-1033.
- Hans-Gill, R. J., Madhu Raka and Ranjeet Sehmi, Estimates On Conjectures of Minkowski and Woods, Indian Jl. Pure Appl. Math., 41(4), (2010), 595-606.
- Hans-Gill, R. J., Madhu Raka and Ranjeet Sehmi, On Conjectures of Minkowski and Woods for n = 8, Acta Arithmetica, 147(4), (2011), 337-385.
- Hans-Gill, R. J., Madhu Raka and Ranjeet Sehmi, Estimates On Conjectures of Minkowski and Woods II, Indian Jl. Pure Appl. Math., 42(5), (2011), 307-333.
- Leetika Kathuria and Madhu Raka, On Conjectures of Minkowski and Woods for n = 9 , Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126 No. 4, (2016), pp. 501-548.
- Leetika Kathuria and Madhu Raka, Rened Estimates on Conjectures of Woods and Minkowski , Jl. Indian Mathematical Society, Vol 83, No.1-2, (2016), 61-85.
- McMullen, C. T., Minkowski's conjecture, well rounded lattices and topological dimension, J. Amer. Math. Soc., 18 (2005), 711-734.
- Regev, O., Shapira, U. and Weiss, B., Counter examples to a conjecture of Woods, Duke Math. J., 166, (2017), 2443-2446.