Decomposable form equations without the finiteness property

Authors:
Zhihua Chen and Min Ru

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1929-1933

MSC (2000):
Primary 11D72

DOI:
https://doi.org/10.1090/S0002-9939-05-07816-0

Published electronically:
January 31, 2005

MathSciNet review:
2137857

Full-text 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 non-zero 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]**Evertse, J.H. and Györy, K.,*Finiteness criteria for decomposable form equations,*Acta Arith.**50**(1988), 357-379. MR**0961695 (90a:11041)****[EG2]**Evertse, J.H. and Györy, K.,*Decomposable form equations*. In: New advances in transcendental theory (ed. by A. Baker), 175-202. 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), 267-275. 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), 237-265. 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), 603-606. 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), 51-65. MR**1423025 (97g:11076)****[RW]**Ru, M. and Wong, P.M.,*Integral points of**hyperplanes in general position*, Invent. Math.**106**(1991), 196-216. MR**1123379 (93f:11056)****[Sch1]**Schmidt, W.M.,*Norm form equations*, Ann. of Math.**(2)96**(1972), 526-551. 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), 455-459. MR**0873474 (88e:32037)****[Th]**Thue, T.,*Über Annäherungswerte algebraischer Zahlen*, J. Reine Angew. Math.,**135**(1909), 284-305.**[V]**Vojta, P.,*Diophantine Approximations and Value Distribution Theory*, Lect. Notes Math. Vol. 1239, Springer, Berlin Heidelberg New York, 1987. MR**0883451 (91k:11049)**

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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:
https://doi.org/10.1090/S0002-9939-05-07816-0

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 MSPF-02G-175.

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2005
American Mathematical Society