Siegel's lemma with additional conditions

✍ Scribed by Lenny Fukshansky

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 157 KB
• Volume: 120
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.11.009

✦ Synopsis

Let K be a number field, and let W be a subspace of K N , N 1. Let V 1 , . . . , V M be subspaces of K N of dimension less than dimension of W . We prove the existence of a point of small height in W \ M i=1 V i , providing an explicit upper bound on the height of such a point in terms of heights of W and V 1 , . . . , V M . Our main tool is a counting estimate we prove for the number of points of a subspace of K N inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.