Baker's theorem

Remarks

The proof extends the method of Gelfond. It builds an auxiliary function in several variables that vanishes to high order at many points, then uses an extrapolation step and an estimate to reach a contradiction. The method also gives an effective lower bound. Let $\Lambda = \beta_0 + \beta_1 \lambda_1 + \dots + \beta_n \lambda_n$ with algebraic $\beta_i$, and suppose $\Lambda \ne 0$. Then $\abs{\Lambda}$ is bounded below by an explicit function of the degrees and heights of the $\alpha_i$ and $\beta_i$. The Gelfond–Schneider theorem is a special case. Another consequence: $e^{\beta_0} \alpha_1^{\beta_1} \cdots \alpha_n^{\beta_n}$ is transcendental for nonzero algebraic $\alpha_i$ and $\beta_i$. For example, $\pi + \log 2$ is transcendental: indeed $\pi = -i \log(-1)$, and $\log(-1) = i\pi$ and $\log 2$ are linearly independent over $\Q$. The hypothesis of independence over $\Q$ is necessary: $\log 4 - 2 \log 2 = 0$ is a relation with algebraic coefficients.

Going deeper

For a student

The effective lower bounds for linear forms in logarithms are the key input for many effective results in Diophantine equations. They give explicit bounds for the solutions of Thue equations and of $S$-unit equations. They were also used in the solution of the class number problem for imaginary quadratic fields of class number one. The theorem is a step toward Schanuel's conjecture, which would give algebraic independence instead of linear independence.