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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the spatial distribution of solutions of decomposable form equations


Authors: G. Everest, I. Gaál, K. Györy and C. Röttger
Journal: Math. Comp. 71 (2002), 633-648
MSC (2000): Primary 11D57, 11Y50
Published electronically: August 3, 2001
MathSciNet review: 1885618
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Abstract:

We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like.


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Additional Information

G. Everest
Affiliation: School of Mathematics, University of East Anglia, Norwich, Norfolk NR4 7TJ, United Kingdom
Email: g.everest@uea.ac.uk

I. Gaál
Affiliation: Institute of Mathematics and Informatics, Lajos Kossuth University, H-4010 Debrecen, Pf 12, Hungary
Email: igaal@math.klte.hu

K. Györy
Affiliation: Institute of Mathematics and Informatics, Lajos Kossuth University, H-4010 Debrecen, Pf 12, Hungary
Email: gyory@math.klte.hu

C. Röttger
Affiliation: School of Mathematics, University of East Anglia, Norwich, Norfolk NR4 7TJ, United Kingdom
Email: C.Rottger@uea.ac.uk

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01353-9
PII: S 0025-5718(01)01353-9
Received by editor(s): July 13, 1999
Received by editor(s) in revised form: June 6, 2000
Published electronically: August 3, 2001
Additional Notes: Röttger’s research was supported by a PhD grant from the UEA. Györy thanks the LMS for a scheme 2 grant at an early stage of this research. Györy and Gaál were supported by the Hungarian Academy of Sciences and by grants 16975, 25157 and 29330 from the Hungarian National Foundation for Scientific Research.
Article copyright: © Copyright 2001 American Mathematical Society