Six exponentials theorem

Last updated September 05, 2024

In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of six exponentials.

Statement

If

x_{1},x_{2},x_{3}

are three complex numbers that are linearly independent over the rational numbers, and

y_{1},y_{2}

are two complex numbers that are also linearly independent over the rational numbers, then at least one of the following numbers is transcendental:

$e^{x_{1}y_{1}},e^{x_{1}y_{2}},e^{x_{2}y_{1}},e^{x_{2}y_{2}},e^{x_{3}y_{1}},e^{x_{3}y_{2}}.$

The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:

{\mathcal {L}}=\{\lambda \in \mathbb {C} \,:\,e^{\lambda }\in {\overline {\mathbb {Q} }}\}.

The theorem then says that if $\lambda _{ij}$ are elements of L for $i=1,2,j=1,2,3$ such that $\lambda _{11},\lambda _{12},\lambda _{13}$ are linearly independent over the rational numbers, and λ₁₁ and λ₂₁ are also linearly independent over the rational numbers, then the matrix

M={\begin{pmatrix}\lambda _{11}&\lambda _{12}&\lambda _{13}\\\lambda _{21}&\lambda _{22}&\lambda _{23}\end{pmatrix}}

has rank 2.

History

A special case of the result where x₁, x₂, and x₃ are logarithms of positive integers, y₁ = 1, and y₂ is real, was first mentioned in a paper by Leonidas Alaoglu and Paul Erdős from 1944 in which they try to prove that the ratio of consecutive colossally abundant numbers is always prime. They claimed that Carl Ludwig Siegel knew of a proof of this special case, but it is not recorded.^[1] Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime.

The theorem was first explicitly stated and proved in its complete form independently by Serge Lang ^[2] and Kanakanahalli Ramachandra ^[3] in the 1960s.

Five exponentials theorem

A stronger, related result is the five exponentials theorem,^[4] which is as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:

e^{x_{1}y_{1}},e^{x_{1}y_{2}},e^{x_{2}y_{1}},e^{x_{2}y_{2}},e^{\gamma x_{2}/x_{1}}.

This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.

Sharp six exponentials theorem

Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem.^[5] This theorem is as follows. Let x₁, x₂, and x₃ be complex numbers that are linearly independent over the rational numbers, and let y₁ and y₂ be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that β_ij are six algebraic numbers for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 such that the following six numbers are algebraic:

e^{x_{1}y_{1}-\beta _{11}},e^{x_{1}y_{2}-\beta _{12}},e^{x_{2}y_{1}-\beta _{21}},e^{x_{2}y_{2}-\beta _{22}},e^{x_{3}y_{1}-\beta _{31}},e^{x_{3}y_{2}-\beta _{32}}.

Then x_i y_j = β_ij for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2. The six exponentials theorem then follows by setting β_ij = 0 for every i and j, while the five exponentials theorem follows by setting x₃ = γ/x₁ and using Baker's theorem to ensure that the x_i are linearly independent.

There is a sharp version of the five exponentials theorem as well, although it as yet unproven so is known as the sharp five exponentials conjecture.^[6] This conjecture implies both the sharp six exponentials theorem and the five exponentials theorem, and is stated as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let α, β₁₁, β₁₂, β₂₁, β₂₂, and γ be six algebraic numbers with γ ≠ 0 such that the following five numbers are algebraic:

e^{x_{1}y_{1}-\beta _{11}},e^{x_{1}y_{2}-\beta _{12}},e^{x_{2}y_{1}-\beta _{21}},e^{x_{2}y_{2}-\beta _{22}},e^{(\gamma x_{2}/x_{1})-\alpha }.

Then x_i y_j = β_ij for 1 ≤ i, j ≤ 2 and γx₂ = αx₁.

A consequence of this conjecture that isn't currently known would be the transcendence of e^π², by setting x₁ = y₁ = β₁₁ = 1, x₂ = y₂ = iπ, and all the other values in the statement to be zero.

Strong six exponentials theorem

The logical implications between the various problems in this circle. Those in red are as yet unproven while those in blue are known results. The top most result refers to that discussed at Baker's theorem, while the four exponentials conjectures are detailed at the four exponentials conjecture article. N-Exponentials Conjecture.png — The logical implications between the various problems in this circle. Those in red are as yet unproven while those in blue are known results. The top most result refers to that discussed at Baker's theorem, while the four exponentials conjectures are detailed at the four exponentials conjecture article.

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.^[7] 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

\beta _{0}+\sum _{i=1}^{n}\beta _{i}\log \alpha _{i},

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.^[8] 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^∗:

x_{1}y_{1},\,x_{1}y_{2},\,x_{2}y_{1},\,x_{2}y_{2},\,x_{1}/x_{2}.

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.

Generalization to commutative group varieties

The exponential function $e z$ uniformizes the exponential map of the multiplicative group $G m$ . Therefore, we can reformulate the six exponential theorem more abstractly as follows:

Let

G = G m \times G m

and take

u : C \to G (C)

to be a non-zero complex-analytic group homomorphism. Define

L

to be the set of complex numbers

l

for which

u (l)

is an algebraic point of

G

. If a minimal generating set of

L

over

Q

has more than two elements then the image

u (C)

is an algebraic subgroup of

G (C)

.

(In order to derive the classical statement, set $u (z) = (e y 1 z; e y 2 z)$ and note that $Q x 1 + Q x 2 + Q x 3$ is a subset of $L$ ).

In this way, the statement of the six exponentials theorem can be generalized to an arbitrary commutative group variety $G$ over the field of algebraic numbers. This generalized six exponential conjecture, however, seems out of scope at the current state of transcendental number theory.

For the special but interesting cases $G = G m \times E$ and $G = E \times E'$ , where $E, E'$ are elliptic curves over the field of algebraic numbers, results towards the generalized six exponential conjecture were proven by Aleksander Momot.^[9] These results involve the exponential function $e z$ and a Weierstrass function $\wp$ resp. two Weierstrass functions $\wp ,\wp '$ with algebraic invariants $g_{2},g_{3},g_{2}',g_{3}'$ , instead of the two exponential functions $e^{y_{1}z},e^{y_{2}z}$ in the classical statement.

Let $G = G m \times E$ and suppose $E$ is not isogenous to a curve over a real field and that $u (C)$ is not an algebraic subgroup of $G (C)$ . Then $L$ is generated over $Q$ either by two elements $x 1, x 2$ , or three elements $x 1, x 2, x 3$ which are not all contained in a real line $R c$ , where $c$ is a non-zero complex number. A similar result is shown for $G = E \times E'$ .^[10]

Notes

↑ Alaoglu and Erdős, (1944), p.455: "Professor Siegel has communicated to us the result that q^x, r^x and s^x can not be simultaneously rational except if x is an integer."
↑ Lang, (1966), chapter 2, section 1.
↑ Ramachandra, (1967/68).
↑ Waldschmidt, (1988), corollary 2.2.
↑ Waldschmidt, (2005), theorem 1.4.
↑ Waldschmidt, (2005), conjecture 1.5
↑ Roy, (1992), section 4, corollary 2.
↑ Waldschmidt, (1988).
↑ Momot, ch. 7
↑ Momot, ch. 7

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References

Alaoglu, Leonidas; Erdős, Paul (1944). "On highly composite and similar numbers". Trans. Amer. Math. Soc. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
Lang, Serge (1966). Introduction to transcendental numbers. Reading, Mass.: Addison-Wesley Publishing Co. MR 0214547.
Momot, Aleksander (2011). "Density of rational points on commutative group varieties and small transcendence degree". arXiv: 1011.3368 [math.NT].
Ramachandra, Kanakanahalli (1967–1968). "Contributions to the theory of transcendental numbers. I, II". Acta Arith. 14: 65–72, 73–88. doi: 10.4064/aa-14-1-65-72 . MR 0224566.
Roy, Damien (1992). "Matrices whose coefficients are linear forms in logarithms". J. Number Theory . 41 (1): 22–47. doi: 10.1016/0022-314x(92)90081-y . MR 1161143.
Waldschmidt, Michel (1988). "On the transcendence methods of Gel'fond and Schneider in several variables". In Baker, Alan (ed.). New advances in transcendence theory. Cambridge University Press. pp. 375–398. MR 0972013.
Waldschmidt, Michel (2005). "Hopf algebras and transcendental numbers". In Aoki, Takashi; Kanemitsu, Shigeru; Nakahara, Mikio; et al. (eds.). Zeta functions, topology, and quantum physics: Papers from the symposium held at Kinki University, Osaka, March 3–6, 2003. Developments in mathematics. Vol. 14. Springer. pp. 197–219. MR 2179279.

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[1] Alaoglu and Erdős, (1944), p.455: "Professor Siegel has communicated to us the result that q^x, r^x and s^x can not be simultaneously rational except if x is an integer."

[2] Lang, (1966), chapter 2, section 1.

[3] Ramachandra, (1967/68).

[4] Waldschmidt, (1988), corollary 2.2.

[5] Waldschmidt, (2005), theorem 1.4.

[6] Waldschmidt, (2005), conjecture 1.5

[7] Roy, (1992), section 4, corollary 2.

[8] Waldschmidt, (1988).

[9] Momot, ch. 7

[10] Momot, ch. 7

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