Waldschmidt 1973: if one column of the four exponentials is algebraic, two of the eight numbers are algebraically independent

Let $x_1, x_2$ be complex numbers linearly independent over $\mathbb{Q}$, and $y_1, y_2$ complex numbers linearly independent over $\mathbb{Q}$. If $e^{x_1y_2}$ and $e^{x_2y_2}$ are algebraic, then two of the eight numbers

are algebraically independent over $\mathbb{Q}$.

Proof. Suppose not. One of $x_1, y_2$ is transcendental by Hermite–Lindemann, since $x_1y_2 \neq 0$ and $e^{x_1y_2}$ is algebraic. The pair lemma Transcendence.isAlgebraic_adjoin_of_not_algebraicIndependent_pair then puts all eight numbers in the algebraic closure of $\mathbb{Q}[t]$ for that transcendental $t$, so they generate a field of transcendence degree at most one. The column construction (FourExp.small_polynomials_of_column, after Waldschmidt's Lemmes 4–7) then produces integer polynomials that are too small at a transcendental point, and Gel'fond's criterion in Waldschmidt's 1971 form (FourExp.transcendence_criterion) makes that point algebraic, a contradiction.

The two algebraic exponentials must share a column. The transcendental ones, $e^{x_1y_1}$ and $e^{x_2y_1}$, are sampled only along the short side of the interpolation grid, where their powers cost degree in the transcendental generator and not height. The four exponentials theorem in transcendence degree one (DiazModulus.four_exponentials_trdeg_one) is the case in which all four exponentials are algebraic.

Novelty. None: this is the Théorème of Waldschmidt (1973), p. 192, restated as Theorem 7.4.1 of Nombres transcendants (1974), Corollary 15.28(c) of Diophantine Approximation on Linear Algebraic Groups (2000) and Corollary 1.2(c) of Roy–Waldschmidt (1997). The paper's note added in proof (p. 202) records that W. D. Brownawell found these results independently. The contribution of this node is the formal proof.