= Schneider lattice proof of the Gelfond–Schneider theorem
{c}
Assume $a$ and $a^b$ algebraic with algebraic irrational $b$. Apply <Siegel lemma> to construct a small-coefficient polynomial $P$ vanishing at the algebraic pairs $(s+tb,a^s(a^b)^t)$ on a large square grid. The function $P(z,e^{\lambda z})$ is a nonzero exponential polynomial. Its many initial zeros make later grid values exponentially tiny, whereas an algebraic norm estimate forbids such tiny nonzero values. Iterating extends the zero grid indefinitely. Its quadratic zero count in disks contradicts the linear zero-count growth of an exponential polynomial.
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