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

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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*

*.*

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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