A Gel’fond Type Criterion in Degree Two Benoit Arbour and Damien Roy


then ξ is algebraic over Q of degree at most n. For example, Brownawell’s version of Gel’fond’s criterion in [1] implies that the above statement holds with any τ > 3n, and the more specific version proved by Davenport and Schmidt as Theorem 2b of [4] shows that it holds with τ = 2n−1. On the other hand, the above application of Dirichlet box principle implies τ ≥ n. So, if we denote by τn the infimum of all admissible values of τ for a fixed n ≥ 1, then we have τ1 = 1 and, in general n ≤ τn ≤ 2n− 1.


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