Decomposable form equations without the finiteness property
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- by Zhihua Chen and Min Ru PDF
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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{equation*} F({\mathbf {x}})=b ~~~\text {in} ~~~~{\mathbf {x}}=(x_{1}, \dots , x_{m})\in R^{m}\end{equation*} 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 $\#{\mathcal {L}} > 2(m-1)$.References
- J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), no. 4, 357–379. MR 961695, DOI 10.4064/aa-50-4-357-379
- 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
- K. Győry, Some applications of decomposable form equations to resultant equations, Colloq. Math. 65 (1993), no. 2, 267–275. MR 1240172, DOI 10.4064/cm-65-2-267-275
- 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
- K. Győry and Min Ru, Integer solutions of a sequence of decomposable form inequalities, Acta Arith. 86 (1998), no. 3, 227–237. MR 1655981, DOI 10.4064/aa-86-3-227-237
- Peter Kiernan, Hyperbolic submanifolds of complex projective space, Proc. Amer. Math. Soc. 22 (1969), 603–606. MR 245828, DOI 10.1090/S0002-9939-1969-0245828-9
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Min Ru and Paul Vojta, Schmidt’s subspace theorem with moving targets, Invent. Math. 127 (1997), no. 1, 51–65. MR 1423025, DOI 10.1007/s002220050114
- Min Ru and Pit-Mann Wong, Integral points of $\textbf {P}^n-\{2n+1\;\textrm {hyperplanes\;in\;general\;position}\}$, Invent. Math. 106 (1991), no. 1, 195–216. MR 1123379, DOI 10.1007/BF01243910
- Wolfgang M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551. MR 314761, DOI 10.2307/1970824
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
- V. E. Snurnitsyn, The complement to $2n$ hyperplanes in $\textbf {C}\textrm {P}^n$ is not hyperbolic, Mat. Zametki 40 (1986), no. 4, 455–459, 552 (Russian). MR 873474
- Thue, T., Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math., 135 (1909), 284-305.
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 1929-1933
- MSC (2000): Primary 11D72
- DOI: https://doi.org/10.1090/S0002-9939-05-07816-0
- MathSciNet review: 2137857