Weak Schanuel conjecture and the four exponentials problem: algebraic independence of logarithms of algebraic numbers, complex and p-adic

This problem concerns the conjecture of algebraic independence of logarithms of algebraic numbers — the weak Schanuel conjecture — together with its most famous consequence, the four exponentials conjecture, in both the complex and the p-adic setting. It asks: if $\lambda_1,\dots,\lambda_n$ are logarithms of algebraic numbers and are linearly independent over the rationals, must they be algebraically independent over the rationals? Already the case $n=2$ is open, and the conjecture at natural generality implies the four exponentials conjecture, which in turn implies that a real number $t$ with $2^t$ and $3^t$ both integers must be a nonnegative integer. The source is Michel Waldschmidt's 2023 survey 'The Four Exponentials Problem and Schanuel's Conjecture', which records all of these questions as open.

Background

A complex number is transcendental if it is not a root of any nonzero polynomial with rational coefficients. Numbers $z_1,\dots,z_n$ are algebraically independent over $\mathbb{Q}$ if no nonzero polynomial with rational coefficients vanishes at $(z_1,\dots,z_n)$ ; this is much stronger than each $z_i$ being transcendental. A logarithm of a nonzero algebraic number $\alpha$ is any complex number $\lambda$ with $e^{\lambda}=\alpha$ ; it is determined only up to addition of multiples of $2\pi i$ . The Lindemann–Weierstrass theorem implies that every nonzero logarithm of a nonzero algebraic number is transcendental (so $\pi$ and $\log 2$ are transcendental), but says nothing about algebraic independence of two such logarithms. The Gelfond–Schneider theorem implies that the quotient of two $\mathbb{Q}$ -linearly independent logarithms of algebraic numbers is transcendental, and Baker's theorem on linear forms in logarithms (Baker 1966) implies that $\mathbb{Q}$ -linearly independent logarithms of algebraic numbers are even linearly independent over the field of all algebraic numbers. Beyond these linear results nothing is known in the direction of algebraic independence: no explicit pair of $\mathbb{Q}$ -linearly independent logarithms of algebraic numbers — not even $\log 2$ and $\log 3$ — is known to be algebraically independent. The classical six exponentials theorem (published proofs by Lang 1966 and Ramachandra 1968) is the strongest general statement of this type: if $x_1,x_2,x_3$ are linearly independent over $\mathbb{Q}$ and $y_1,y_2$ are linearly independent over $\mathbb{Q}$ , then at least one of the six numbers $e^{x_i y_j}$ is transcendental. The boundary case with two $x$ 's and two $y$ 's is the four exponentials conjecture and is open: it asserts that for $\mathbb{Q}$ -linearly independent pairs $(x_1,x_2)$ and $(y_1,y_2)$ , at least one of the four numbers $e^{x_1y_1},e^{x_1y_2},e^{x_2y_1},e^{x_2y_2}$ is transcendental. The dimension-counting at the heart of all known proof methods degenerates exactly at this $(2,2)$ case. Schanuel's conjecture itself, the central open problem of the field, asserts more: if complex numbers $x_1,\dots,x_n$ are linearly independent over $\mathbb{Q}$ , then the field generated over $\mathbb{Q}$ by the $x_i$ together with their exponentials $e^{x_i}$ has transcendence degree at least $n$ over $\mathbb{Q}$ — that is, it contains at least $n$ algebraically independent elements. Its restriction to tuples that are logarithms of algebraic numbers is the umbrella conjecture of this problem, the weak Schanuel conjecture: $\mathbb{Q}$ -linearly independent logarithms of algebraic numbers are algebraically independent; for such a tuple the exponentials $e^{\lambda_i}$ are algebraic, so the transcendence-degree lower bound collapses to algebraic independence of the $\lambda_i$ alone. The weak Schanuel conjecture implies the four exponentials conjecture — in the source's words, the latter is 'a very special case' of the former. For suppose all four $e^{x_iy_j}$ were algebraic; then the four products $\lambda_{ij}=x_iy_j$ would be logarithms of algebraic numbers satisfying the multiplicative relation $\lambda_{11}\lambda_{22}=\lambda_{12}\lambda_{21}$ . Take a maximal $\mathbb{Q}$ -linearly independent subcollection of the four products — it has at least two elements, since the entries of any row or column are $\mathbb{Q}$ -linearly independent — and expand each remaining product as a rational linear combination of it; the multiplicative relation becomes a nonzero polynomial relation among the subcollection (when the subcollection is the two entries of the first row, the polynomial could vanish identically only if $x_2$ were a rational multiple of $x_1$ , which the hypothesis forbids), contradicting weak Schanuel. These questions have an exact $p$ -adic analogue. The field $\mathbb{C}_p$ is the completion of an algebraic closure of the field $\mathbb{Q}_p$ of $p$ -adic numbers; the $p$ -adic logarithm is defined on units congruent to $1$ modulo $p$ and extended to all of $\mathbb{C}_p^{\times}$ by continuity with $\log_p(p)=0$ (the standard Iwasawa extension, which has a nontrivial kernel), and it plays the role of the complex logarithm in $p$ -adic transcendence theory. Leopoldt's conjecture (Leopoldt 1962) asserts that the $p$ -adic rank of the units of a number field equals its archimedean rank, equivalently that a certain $p$ -adic regulator does not vanish; it is open for general number fields, known for abelian number fields by work of Brumer (1967) using Baker's theorem. Waldschmidt's 2023 survey takes Leopoldt's conjecture as the leitmotiv for the $p$ -adic four exponentials and weak Schanuel questions, which remain open in the $p$ -adic setting just as in the complex one (Waldschmidt 2023, Waldschmidt 2000).

Problem Statement

Prove or refute the weak Schanuel conjecture — the conjecture of algebraic independence of logarithms of algebraic numbers — at its natural generality, in both the complex and the p-adic setting. Complex statement: for every $n\ge 2$ and all logarithms $\lambda_1,\dots,\lambda_n$ of nonzero algebraic numbers (i.e. complex numbers with $e^{\lambda_i}$ algebraic), if $\lambda_1,\dots,\lambda_n$ are linearly independent over $\mathbb{Q}$ then they are algebraically independent over $\mathbb{Q}$ . p-adic statement: the same assertion for $p$ -adic logarithms of algebraic numbers — for a rational prime $p$ and algebraic numbers $\alpha_1,\dots,\alpha_n\in\mathbb{C}_p^{\times}$ , the values $\lambda_i=\log_p(\alpha_i)$ of the $p$ -adic logarithm (under its standard extension to all of $\mathbb{C}_p^{\times}$ , as described in the background) must be algebraically independent over $\mathbb{Q}$ whenever they are linearly independent over $\mathbb{Q}$ . The two settings are parallel forms of one objective — establish (or disprove) algebraic independence of $\mathbb{Q}$ -linearly independent logarithms of algebraic numbers — and an answer may resolve either setting separately; no restriction on the number $n$ of logarithms, on the algebraic inputs, or on the number field from which they come may be imposed. In particular, the complex statement implies the four exponentials conjecture: for $\mathbb{Q}$ -linearly independent pairs $(x_1,x_2)$ and $(y_1,y_2)$ of complex numbers, at least one of the four numbers $e^{x_1y_1},e^{x_1y_2},e^{x_2y_1},e^{x_2y_2}$ is transcendental; this open conjecture in turn implies a positive answer to the emblematic question recorded in the source: if a real number $t$ satisfies that $2^t$ and $3^t$ are both integers, must $t$ be a nonnegative integer? The objective is the conjecture at its natural generality; the four exponentials conjecture, finitely many special cases, and results restricted to classes of algebraic inputs are proper sub-cases of it, not the objective.

References

Michel Waldschmidt, The Four Exponentials Problem and Schanuel's Conjecture, in: Mathematics Going Forward (J.-M. Morel, B. Teissier, eds.), Lecture Notes in Mathematics 2313, Springer, Cham, 2023, pp. 579–592. DOI: 10.1007/978-3-031-12244-6_39, https://doi.org/10.1007/978-3-031-12244-6_39
James Ax, On Schanuel's Conjectures, Annals of Mathematics (2) 93 (1971), 252–268. DOI: 10.2307/1970774, https://doi.org/10.2307/1970774
A. Baker, Linear forms in the logarithms of algebraic numbers, Mathematika 13 (1966), 204–216. DOI: 10.1112/S0025579300003971, https://doi.org/10.1112/S0025579300003971
Armand Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121–124. DOI: 10.1112/s0025579300003703, https://doi.org/10.1112/s0025579300003703
Heinrich-Wolfgang Leopoldt, Zur Arithmetik in abelschen Zahlkörpern, Journal für die reine und angewandte Mathematik 209 (1962), 54–71. DOI: 10.1515/crll.1962.209.54, https://doi.org/10.1515/crll.1962.209.54
K. Ramachandra, Contributions to the theory of transcendental numbers (I), Acta Arithmetica 14 (1968), 65–72. DOI: 10.4064/aa-14-1-65-72, https://doi.org/10.4064/aa-14-1-65-72
Serge Lang, Introduction to Transcendental Numbers, Addison-Wesley, Reading, MA, 1966 (reprinted in Serge Lang, Collected Papers, Volume I, Springer, 2000; ISBN 978-1-4612-2118-0). DOI of reprint chapter: 10.1007/978-1-4612-2118-0_37, https://doi.org/10.1007/978-1-4612-2118-0_37
Michel Waldschmidt, Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables, Grundlehren der mathematischen Wissenschaften 326, Springer, Berlin, 2000. DOI: 10.1007/978-3-662-11569-5, https://doi.org/10.1007/978-3-662-11569-5
Samit Dasgupta, Ranks of matrices of logarithms of algebraic numbers, I: The theorems of Baker and Waldschmidt–Masser, Essential Number Theory 2 (2023), 93–138. DOI: 10.2140/ent.2023.2.93, https://doi.org/10.2140/ent.2023.2.93
Alexandre Maksoud, On the rank of Leopoldt's and Gross's regulator maps, Documenta Mathematica 28 (2023), 1441–1471 (preprint arXiv:2201.08203). DOI: 10.4171/dm/935, https://doi.org/10.4171/dm/935
M. Emsalem, H. H. Kisilevsky, D. B. Wales, Indépendance linéaire sur Q̄ de logarithmes p-adiques de nombres algébriques et rang p-adique du groupe des unités d'un corps de nombres, Journal of Number Theory 19 (1984), 384–391. DOI: 10.1016/0022-314X(84)90079-9, https://doi.org/10.1016/0022-314X(84)90079-9
Michel Laurent, Rang p-adique d'unités et action de groupes, Journal für die reine und angewandte Mathematik 399 (1989), 81–108. DOI: 10.1515/crll.1989.399.81, https://doi.org/10.1515/crll.1989.399.81
Laurent Habsieger, A few remarks related to the four exponentials conjecture, Publicationes Mathematicae Debrecen 70 (2007), 179–194. DOI: 10.5486/pmd.2007.3505, https://doi.org/10.5486/pmd.2007.3505
Lee A. Butler, A Diophantine approach to the three and four exponentials conjectures, The Ramanujan Journal 42 (2017), 199–221. DOI: 10.1007/s11139-015-9728-2, https://doi.org/10.1007/s11139-015-9728-2
Guy Diaz, Produits et quotients de combinaisons linéaires de logarithmes de nombres algébriques : conjectures et résultats partiels, Journal de théorie des nombres de Bordeaux 19 (2007), 373–391. DOI: 10.5802/jtnb.592, https://doi.org/10.5802/jtnb.592
B. Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic 132 (2005), 67–95. DOI: 10.1016/j.apal.2004.07.001, https://doi.org/10.1016/j.apal.2004.07.001