Decomposable form equations without the finiteness property
Authors:
Zhihua Chen and Min Ru
Journal:
Proc. Amer. Math. Soc. 133 (2005), 19291933
MSC (2000):
Primary 11D72
Published electronically:
January 31, 2005
MathSciNet review:
2137857
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a finitely generated (but not necessarily algebraic) extension field of . Let be a form (homogeneous polynomial) in variables with coefficients in , and suppose that is decomposable (i.e., that it factorizes into linear factors over some finite extension of ). We say that has the finiteness property over if for every (here denotes the set of nonzero elements in ) and for every subring of which is finitely generated over , the equation
has only finitely many solutions. This paper proves the following result: Let be a decomposable form in variables with coefficients in , which factorizes into linear factors over . Let denote a maximal set of pairwise linearly independent linear factors of . If has the finiteness property over , then .
 [EG1]
J.H.
Evertse and K.
Győry, Finiteness criteria for decomposable form
equations, Acta Arith. 50 (1988), no. 4,
357–379. MR
961695 (90a:11041)
 [EG2]
J.H.
Evertse and K.
Győry, Decomposable form equations, New advances in
transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge,
1988, pp. 175–202. MR 971999
(89i:11042)
 [EG3]
K.
Győry, Some applications of decomposable form equations to
resultant equations, Colloq. Math. 65 (1993),
no. 2, 267–275. MR 1240172
(94k:11036)
 [G]
K.
Győry, On the distribution of solutions of decomposable form
equations, Number theory in progress, Vol. 1
(ZakopaneKościelisko, 1997) de Gruyter, Berlin, 1999,
pp. 237–265. MR 1689508
(2000e:11036)
 [GR]
K.
Győry and Min
Ru, Integer solutions of a sequence of decomposable form
inequalities, Acta Arith. 86 (1998), no. 3,
227–237. MR 1655981
(2000a:11044)
 [K]
Peter
Kiernan, Hyperbolic submanifolds of complex
projective space, Proc. Amer. Math. Soc. 22 (1969), 603–606.
MR
0245828 (39 #7134), http://dx.doi.org/10.1090/S00029939196902458289
 [L]
Serge
Lang, Fundamentals of Diophantine geometry, SpringerVerlag,
New York, 1983. MR 715605
(85j:11005)
 [RV]
Min
Ru and Paul
Vojta, Schmidt’s subspace theorem with moving targets,
Invent. Math. 127 (1997), no. 1, 51–65. MR 1423025
(97g:11076), http://dx.doi.org/10.1007/s002220050114
 [RW]
Min
Ru and PitMann
Wong, Integral points of
𝑃ⁿ{2𝑛+1ℎ𝑦𝑝𝑒𝑟𝑝𝑙𝑎𝑛𝑒𝑠𝑖𝑛𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛},
Invent. Math. 106 (1991), no. 1, 195–216. MR 1123379
(93f:11056), http://dx.doi.org/10.1007/BF01243910
 [Sch1]
Wolfgang
M. Schmidt, Norm form equations, Ann. of Math. (2)
96 (1972), 526–551. MR 0314761
(47 #3313)
 [Sch2]
Wolfgang
M. Schmidt, Diophantine approximation, Lecture Notes in
Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
(81j:10038)
 [Sn]
V.
E. Snurnitsyn, The complement to 2𝑛 hyperplanes in
𝐶𝑃ⁿ is not hyperbolic, Mat. Zametki
40 (1986), no. 4, 455–459, 552 (Russian). MR 873474
(88e:32037)
 [Th]
Thue, T., Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math., 135 (1909), 284305.
 [V]
Paul
Vojta, Diophantine approximations and value distribution
theory, Lecture Notes in Mathematics, vol. 1239, SpringerVerlag,
Berlin, 1987. MR
883451 (91k:11049)
 [EG1]
 Evertse, J.H. and Györy, K., Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), 357379. MR 0961695 (90a:11041)
 [EG2]
 Evertse, J.H. and Györy, K., Decomposable form equations. In: New advances in transcendental theory (ed. by A. Baker), 175202. Cambridge Univ. Press, Cambridge 1988. MR 0971999 (89i:11042)
 [EG3]
 Evertse, J.H. and Györy, K., Some applications of decomposable equations to resultant equations, Colloq. Math. 65 (1993), 267275. MR 1240172 (94k:11036)
 [G]
 Györy, K., On the distribution of solutions of decomposable form equations, Number Theory in Progress, Walter de Gruyter, Berlin, New York 1 (1999), 237265. MR 1689508 (2000e:11036)
 [GR]
 Györy, K. and Ru, M., Integer solutions of a sequence of decomposable form inequalities, Acta Arith. (1998). MR 1655981 (2000a:11044)
 [K]
 Kiernan, P., Hyperbolic submanifolds of complex projective space, Proc. Amer. Math. Soc. 3 (1968), 603606. MR 0245828 (39:7134)
 [L]
 Lang, S., Fundamentals of Diophantine Geometry, Springer, Berlin Heidelberg New York, 1983. MR 0715605 (85j:11005)
 [RV]
 Ru, M. and Vojta, P., Schmidt's subspace theorem with moving targets, Invent. Math. 127 (1997), 5165. MR 1423025 (97g:11076)
 [RW]
 Ru, M. and Wong, P.M., Integral points of hyperplanes in general position, Invent. Math. 106 (1991), 196216. MR 1123379 (93f:11056)
 [Sch1]
 Schmidt, W.M., Norm form equations, Ann. of Math. (2)96 (1972), 526551. MR 0314761 (47:3313)
 [Sch2]
 Schmidt, W.M., Diophantine approximations, Lect. Notes Math. Vol 785, Springer, Berlin Heidelberg New York, 1980. MR 0568710 (81j:10038)
 [Sn]
 Snurnitsyn, V.E., The complement of hyperplanes is not hyperbolic, Mat. Zametki 40 (1986), 455459. MR 0873474 (88e:32037)
 [Th]
 Thue, T., Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math., 135 (1909), 284305.
 [V]
 Vojta, P., Diophantine Approximations and Value Distribution Theory, Lect. Notes Math. Vol. 1239, Springer, Berlin Heidelberg New York, 1987. MR 0883451 (91k:11049)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11D72
Retrieve articles in all journals
with MSC (2000):
11D72
Additional Information
Zhihua Chen
Affiliation:
Department of Mathematics, Tongji University, Shanghai, People’s Republic of China
Email:
zzzhhc@tongji.edu.cn
Min Ru
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204
Email:
minru@math.uh.edu
DOI:
http://dx.doi.org/10.1090/S0002993905078160
PII:
S 00029939(05)078160
Received by editor(s):
December 5, 2003
Received by editor(s) in revised form:
March 18, 2004
Published electronically:
January 31, 2005
Additional Notes:
The first author was supported by NSFC number 10271089. The second author was supported in part by NSA under grant number MSPF02G175.
Communicated by:
WenChing Winnie Li
Article copyright:
© Copyright 2005 American Mathematical Society
