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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: January 31, 2005
MathSciNet review: 2137857
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Abstract: Let $K$ be a finitely generated (but not necessarily algebraic) extension field of ${\mathbb{Q}}$. Let $F({\mathbf{X}})=F(X_{1}, \dots , X_{m})$be a form (homogeneous polynomial) in $m \ge 2$ variables with coefficients in $K$, and suppose that $F$ is decomposable (i.e., that it factorizes into linear factors over some finite extension of $K$). We say that $F$ has the finiteness property over $K$ if for every $b \in K^{*}$ (here $K^{*}$ denotes the set of non-zero elements in $K$) and for every subring $R$ of $K$ which is finitely generated over ${\mathbb{Z}}$, the equation

\begin{displaymath}F({\mathbf{x}})=b ~~~\text{in} ~~~~{\mathbf{x}}=(x_{1}, \dots , x_{m})\in R^{m}\end{displaymath}

has only finitely many solutions. This paper proves the following result: Let $F$ be a decomposable form in $m \ge 2$ variables with coefficients in $K$, which factorizes into linear factors over $K$. Let ${\mathcal{L}}$ denote a maximal set of pairwise linearly independent linear factors of $F$. If $F$ has the finiteness property over $K$, then $\char93 {\mathcal{L}} > 2(m-1)$.


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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: http://dx.doi.org/10.1090/S0002-9939-05-07816-0
PII: S 0002-9939(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