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]**J.-H. Evertse and K. Győry,*Finiteness criteria for decomposable form equations*, Acta Arith.**50**(1988), no. 4, 357–379. MR**961695****[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****[EG3]**K. Győry,*Some applications of decomposable form equations to resultant equations*, Colloq. Math.**65**(1993), no. 2, 267–275. MR**1240172****[G]**K. Győry,*On the distribution of solutions of decomposable form equations*, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 237–265. MR**1689508****[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****[K]**Peter Kiernan,*Hyperbolic submanifolds of complex projective space*, Proc. Amer. Math. Soc.**22**(1969), 603–606. MR**0245828**, https://doi.org/10.1090/S0002-9939-1969-0245828-9**[L]**Serge Lang,*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605****[RV]**Min Ru and Paul Vojta,*Schmidt’s subspace theorem with moving targets*, Invent. Math.**127**(1997), no. 1, 51–65. MR**1423025**, https://doi.org/10.1007/s002220050114**[RW]**Min Ru and Pit-Mann Wong,*Integral points of 𝑃ⁿ-{2𝑛+1ℎ𝑦𝑝𝑒𝑟𝑝𝑙𝑎𝑛𝑒𝑠𝑖𝑛𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛}*, Invent. Math.**106**(1991), no. 1, 195–216. MR**1123379**, https://doi.org/10.1007/BF01243910**[Sch1]**Wolfgang M. Schmidt,*Norm form equations*, Ann. of Math. (2)**96**(1972), 526–551. MR**0314761**, https://doi.org/10.2307/1970824**[Sch2]**Wolfgang M. Schmidt,*Diophantine approximation*, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR**568710****[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****[Th]**Thue, T.,*Über Annäherungswerte algebraischer Zahlen*, J. Reine Angew. Math.,**135**(1909), 284-305.**[V]**Paul Vojta,*Diophantine approximations and value distribution theory*, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR**883451**

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:
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