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 | References | Similar Articles | Additional Information

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.

**[E1]**G. R. Everest,*Angular distribution of units in abelian group rings—an application to Galois-module theory*, J. Reine Angew. Math.**375/376**(1987), 24–41. MR**882289**, 10.1515/crll.1987.375-376.24**[E2]**G. R. Everest,*Units in abelian group rings and meromorphic functions*, Illinois J. Math.**33**(1989), no. 4, 542–553. MR**1007893****[EG]**G. R. Everest and K. Győry,*Counting solutions of decomposable form equations*, Acta Arith.**79**(1997), no. 2, 173–191. MR**1438600****[EvG]**J.-H. Evertse and K. Győry,*The number of families of solutions of decomposable form equations*, Acta Arith.**80**(1997), no. 4, 367–394. MR**1450929****[G]**K. Győry,*On the numbers of families of solutions of systems of decomposable form equations*, Publ. Math. Debrecen**42**(1993), no. 1-2, 65–101. MR**1208854****[K]**Gregory Karpilovsky,*Unit groups of classical rings*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. MR**978631****[Ka]**M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger,*KANT V4*, J. Symbolic Comput.**24**(1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR**1484479**, 10.1006/jsco.1996.0126**[KN]**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0419394****[Sc1]**Wolfgang M. Schmidt,*Norm form equations*, Ann. of Math. (2)**96**(1972), 526–551. MR**0314761****[Sc2]**Wolfgang M. Schmidt,*Diophantine approximation*, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR**568710****[Sc3]**Wolfgang M. Schmidt,*Diophantine approximations and Diophantine equations*, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR**1176315****[Se]**Sudarshan K. Sehgal,*Topics in group rings*, Monographs and Textbooks in Pure and Applied Math., vol. 50, Marcel Dekker, Inc., New York, 1978. MR**508515**

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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:
https://doi.org/10.1090/S0025-5718-01-01353-9

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