Worked solution: Gelfond–Schneider transcendence proof via auxiliary functions (1934)

Frequently asked questions

Why go to the trouble of forcing

\Phi

to vanish to high order, rather than just at one point?

Vanishing to high order at each point packs far more 'zero conditions' into the same set of points, which is what lets the later analytic estimate (via Jensen's or Schwarz's formula) become so strong that it beats any possible algebraic lower bound — a single simple zero at each point would not be nearly powerful enough.

Source

Aleksandr Gelfond and Theodor Schneider (independently), 1934

Sur le septième problème de Hilbert

Step 3 of 7: Building the auxiliary function via Siegel's lemma

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In plain words

The heart of the proof is a single cleverly built function $\Phi(z)$ , a combination of $e^{iz}$ and $e^{j\beta z}$ for many pairs of whole numbers $(i,j)$ , with coefficients $c_{ij}$ chosen from the number field $K$ . There are more coefficients (roughly $J^2$ of them, for a large parameter $J$ ) than there are conditions demanding $\Phi$ and its low-order derivatives vanish at a handful of chosen points — so a nontrivial solution is guaranteed to exist by a simple counting/pigeonhole argument, without ever writing the coefficients down explicitly.

This counting trick — turning an existence question about numbers into a linear algebra problem with more unknowns than equations — is called Siegel's lemma, and it lets one build $\Phi$ so that it (and many derivatives) vanish exactly at the points $z_0=0,z_1,\dots,z_m$ where $z_j=j\log\alpha$ , keeping the coefficients $c_{ij}$ from growing too large in size.

\Phi(z) = \sum_{i=0}^{J}\sum_{j=0}^{J} c_{ij}\, e^{iz}\, e^{j\beta z}

Detailed analysis

Siegel's lemma states: given $r$ homogeneous linear equations in $n>r$ unknowns with integer coefficients of absolute value at most $A$ , there is a nontrivial integer solution of absolute value at most $2(2nA)^{r/(n-r)}$ (Siu, 'Siegel's Lemma'; proof by a pigeonhole/box argument comparing the number of lattice points in two boxes under the linear map). An extension handles equations with coefficients in the ring of algebraic integers of $K$ , at the cost of constants depending only on $K$ .

Fix a large parameter $L$ , let $J$ be roughly $\sqrt{L\log L}$ , and choose $m+1$ points $z_0=0,z_1,\dots,z_m\in\{j\log\alpha\}$ . Using the differential relations $\frac{d}{dz}e^{iz}e^{j\beta z}=(i+j\beta)e^{iz}e^{j\beta z}$ — a polynomial relation of the type needed for Siegel's lemma to control derivative sizes — one constructs a polynomial $G(X,Y)$ of degree at most $J$ in each variable, with coefficients in $K$ , such that $\Phi(z)=G(e^z,e^{\beta z})$ vanishes to order at least $L$ at each of $z_0,\dots,z_m$ (Siu, 'Proof of Main Theorem'). Siegel's lemma guarantees such a nonzero $G$ exists with coefficients of controlled size (bounded 'size', roughly $\lesssim L$ ), because the number of unknowns ( $\approx J^2$ ) exceeds the number of vanishing conditions ( $\approx mL$ ) for suitable choice of $J,L,m$ .

Crucially, evaluating $\Phi$ and its derivatives at the points $z_j=j\log\alpha$ produces algebraic numbers lying in the fixed field $K$ : because $e^{z_j}=\alpha^j\in K$ and $e^{\beta z_j}=(\alpha^\beta)^j=\gamma^j\in K$ (using the assumed algebraicity of $\gamma$ !), every derivative of $\Phi$ at $z_j$ is an explicit polynomial expression in $\alpha,\beta,\gamma\in K$ . This is exactly where the auxiliary function begins to 'see' the assumption that $\gamma=\alpha^\beta$ is algebraic.

Terms in this step

Siegel's lemma: A pigeonhole-principle result guaranteeing a nontrivial small integer (or algebraic-integer) solution to a system of linear equations whenever there are more unknowns than equations — the standard tool for building auxiliary functions in transcendence theory without needing to compute their coefficients explicitly.