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In fact Schneider showed that αω + βη is transcendental where η is the quasi-period associated with a primitive ω, that is η = 2ζ 12 ω where ζ is the Weierstrass ζ -function given by ζ (z) = −℘ (z) and α, β are algebraic numbers not both zero. As he 30 Logarithmic forms observed, this implies that the ellipse y2 x2 + =1 a2 b2 has a transcendental circumference if a, b are algebraic. 2 due to Schneider is the transcendence of the elliptic modular function j(z) where z is algebraic with positive imaginary part other than a quadratic irrational; in the exceptional case, j(z) is algebraic with degree given by the class number of the quadratic field.

N are linearly independent over Q. 4 include as special cases the classical results of Hermite and Lindemann on the transcendence of e and π , as well as the Gelfond–Schneider theorem, and are plainly of interest from the point of view of the theory of transcendental numbers. However, of critical importance in connection with applications to the solution of Diophantine problems is the fact that the method of proof is effective and yields a sufficiently strong lower bound for | |. After initial work in this context by Baker [15, III] yielding an estimate for = 0 of the κ form | | e−(log B) where κ > n + 1, Feldman [95, 98] in 1971 improved the result to give the following theorem.

Mn−1 with m0 + · · · + mn−1 < k we have | f (z)| < C3hk+L|z| . Further, if l is an integer with 0 ≤ l ≤ hk 8n , then either 1 | f (l)| < B− 2 C or | f (l)| > C4−hk−Ll . Proof. 6 and straightforward bounds for the αj j . For the second part we note that if we replace the quantity β β n−1 αn = α1 1 · · · αn−1 which occurs in f (l) by αn so that the expression γ l γ λ n−1 n−1 = α1λ1 l · · · αn−1 αn α1 1 · · · αn−1 l l λn l becomes α1λ1 l · · · αnλn l , then, apart from a factor m0 ! 1). 4 we see that log αn − log αn < B−C for some value of the second logarithm and some C which we suppose is sufficiently large in terms of k.