Six Exponentials Theorem - Strong Six Exponentials Theorem

A further strengthening of the theorems and conjectures in this area are the strong versions. The strong six exponentials theorem is a result proved by Damien Roy that implies the sharp six exponentials theorem. This result concerns the vector space over the algebraic numbers generated by 1 and all logarithms of algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form

for some n ≥ 0, where all the β_i and α_i are algebraic and every branch of the logarithm is considered. The strong six exponentials theorem then says that if x₁, x₂, and x₃ are complex numbers that are linearly independent over the algebraic numbers, and if y₁ and y₂ are a pair of complex numbers that are also linearly independent over the algebraic numbers then at least one of the six numbers x_i y_j for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 is not in L∗. This is stronger than the standard six exponentials theorem which says that one of these six numbers is not simply the logarithm of an algebraic number.

There is also a strong five exponentials conjecture formulated by Michel Waldschmidt It would imply both, the strong six exponentials theorem and the sharp five exponentials conjecture. This conjecture claims that if x₁, x₂ and y₁, y₂ are two pairs of complex numbers, with each pair being linearly independent over the algebraic numbers, then at least one of the following five numbers is not in L∗:

All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.