5 The six exponentials theorem

Schneider’s method in the form of Lang and Ramachandra, relativised to a fixed number field: an auxiliary function built by Siegel’s lemma, a descent that extends its zeros over a lattice, and a zero estimate.

In this chapter \(x=(x_1,\dots ,x_d)\) and \(y=(y_1,\dots ,y_l)\) are families of complex numbers, each linearly independent over \(\mathbb {Q}\). For \(\lambda \in \mathbb {N}^{d}\) and \(m\in \mathbb {N}^{l}\) write \(\langle \lambda ,x\rangle =\sum _i\lambda _ix_i\) and \(\langle m,y\rangle =\sum _jm_jy_j\). For \(L\in \mathbb {N}\) and integers \(p_\lambda \) the auxiliary function is

\[ F_p(z)=\sum _{\lambda \in \{ 0,\dots ,L-1\} ^{d}}p_\lambda \, e^{\langle \lambda ,x\rangle z}. \]

A number field \(K\) is a subfield of \(\mathbb {C}\) of finite degree over \(\mathbb {Q}\).

Lemma 5.1 The auxiliary function

✓

Let \(d+l\lt dl\), and let \(K\) be a number field containing every \(e^{x_iy_j}\). There is \(c\gt 0\) such that for every large \(M\) there are an integer \(L\ge 1\) with \(L^{d}\le cM^{l}\) and integers \(p_\lambda \) with \(|p_\lambda |\le e^{cLM}\), not all zero for \(\lambda \in \{ 0,\dots ,L-1\} ^{d}\), such that \(F_p(\langle m,y\rangle )=0\) for every \(m\in \mathbb {N}^{l}\) with \(m_j\lt M\) for all \(j\).

Source: [ Lang66 , Ch. II ] , [ Ram68 ] ; see [ Wal09 , § 3.4 ] .

Proof ▶

At the lattice points the exponentials are products of powers of the \(e^{x_iy_j}\), so the conditions form a linear system over \(K\): \(M^{l}\) conditions on \(L^{d}\) unknowns, and \(d+l\lt dl\) leaves room for Siegel’s lemma (Lemma 2.5), with the denominators and houses of Lemma 2.2.

Lemma 5.2 The descent step

✓

Let \(d+l\lt dl\), let \(K\) be a number field containing every \(e^{x_iy_j}\), and let \(c\gt 0\). There is \(M_0\) such that for all \(M\ge M_0\), all \(L\ge 1\) with \(L^{d}\le cM^{l}\), all integers \(p_\lambda \) with \(|p_\lambda |\le e^{cLM}\) and all \(N\ge M\): if \(F_p(\langle m,y\rangle )=0\) for every \(m\in \mathbb {N}^{l}\) with all \(m_j\lt N\), then \(F_p(\langle m,y\rangle )=0\) for every \(m\in \mathbb {N}^{l}\) with all \(m_j\lt N+1\).

Source: [ Lang66 , Ch. II ] , [ Ram68 ] ; see [ Wal09 , § 3.4 ] .

Proof ▶

At a new lattice point \(F_p\) is small, by Lemma 2.9 applied to the zeros already known. Its value there is an element of \(K\) with controlled denominator and house (Lemma 2.2), so if it were non-zero Lemma 2.1 would bound it from below. The two bounds are incompatible because \(dl\gt d+l\).

Lemma 5.3 Vanishing on the whole lattice

✓

Let \(l\ge 2\), \(L\in \mathbb {N}\) and \(p_\lambda \) integers. If \(F_p(\langle m,y\rangle )=0\) for every \(m\in \mathbb {N}^{l}\), then \(p_\lambda =0\) for every \(\lambda \in \{ 0,\dots ,L-1\} ^{d}\).

Source: [ Lang66 , Ch. II ] , [ Ram68 ] ; see [ Wal09 , § 3.4 ] .

Proof ▶

The exponents \(\langle \lambda ,x\rangle \) are pairwise distinct, and as \(l\ge 2\) and the \(y_j\) are independent, \(\lambda \mapsto (e^{\langle \lambda ,x\rangle y_j})_j\) is injective; Dedekind’s independence of characters concludes.

Theorem 5.4 Six exponentials over a number field

✓

Let \(d,l\in \mathbb {N}\) with \(d+l\lt dl\), let \(x_1,\dots ,x_d\) and \(y_1,\dots ,y_l\) be complex numbers, each family linearly independent over \(\mathbb {Q}\), and let \(K\) be a number field. Then some \(e^{x_iy_j}\) does not lie in \(K\).

Source: [ Lang66 , Ch. II ] , [ Ram68 ] ; see [ Wal00 , Theorem 1.12 ] .

Proof ▶

Otherwise Lemma 5.1 gives an auxiliary function, Lemma 5.2 makes it vanish on the whole lattice by induction, and Lemma 5.3 makes all its coefficients zero (\(d+l\lt dl\) forces \(l\ge 2\)), a contradiction.

Theorem 5.5 The six exponentials theorem

✓

Let \(x_1,x_2\) be complex numbers linearly independent over \(\mathbb {Q}\), and let \(y_1,y_2,y_3\) be complex numbers linearly independent over \(\mathbb {Q}\). Then at least one of the six numbers \(e^{x_iy_j}\) (\(i=1,2\), \(j=1,2,3\)) is transcendental.

Source: Lang [ Lang66 , Ch. II ] , Ramachandra [ Ram68 ] ; see [ Wal00 , § 1.3, Theorem 1.12 ] .

Proof ▶

Finitely many algebraic numbers lie in one number field; apply Theorem 5.4 with \((d,l)=(2,3)\).