Integer solutions to decomposable form inequalities

✍ Scribed by Zhihua Chen; Min Ru

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 136 KB
• Volume: 115
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.10.004

✦ Synopsis

This paper obtains a result on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F (X 1 , ..., X m ) be a non-degenerate decomposable form with coefficients in k. We prove that, for every finite set of places S of k containing the archimedean places of k, for each real number < 1 m-1 and for each constant c > 0, the inequality 0 < ∈S F (x 1 , ..., x m ) cH S (x 1 , ..., x m ) in (x 1 , ..., x m ) ∈ O m S .

(1)

has only finitely many O * S -non-proportional solutions.