Schanuel's conjecture

Schanuel's conjecture (1960s) asserts that for any complex numbers $z_1,\dots,z_n$ that are linearly independent over the rationals, the field generated by these numbers together with their exponentials $e^{z_1},\dots,e^{z_n}$ has transcendence degree at least $n$ over $\mathbb{Q}$ ; equivalently, at least $n$ of the $2n$ numbers $z_1,\dots,z_n,e^{z_1},\dots,e^{z_n}$ are algebraically independent. It is among the most central open problems in transcendental number theory: it subsumes the Lindemann–Weierstrass theorem and would imply, among much else, the algebraic independence of $e$ and $\pi$ . The conjecture remains open in full generality: recent work either proves analogues (differential, model-theoretic, elliptic, or Shimura-theoretic) or uses it conditionally. The problem posed here is to prove or refute the conjecture exactly as stated, with no restriction on $n$ or on the $z_i$ .

Background

Two complex numbers are called algebraically independent (over $\mathbb{Q}$ ) if no nonzero polynomial with rational coefficients vanishes when evaluated at them; more generally, the transcendence degree of a field extension of $\mathbb{Q}$ is the maximal number of algebraically independent elements in it. Numbers $z_1,\dots,z_n$ are $\mathbb{Q}$ -linearly independent if no nontrivial rational linear combination of them is zero. Schanuel's conjecture, formulated by Stephen Schanuel in the 1960s in a conversation with Serge Lang, predicts a universal relation between $\mathbb{Q}$ -linear independence and algebraic independence for the exponential function: if $z_1,\dots,z_n\in\mathbb{C}$ are $\mathbb{Q}$ -linearly independent, then

\operatorname{trdeg}_{\mathbb{Q}}\,\mathbb{Q}(z_1,\dots,z_n,e^{z_1},\dots,e^{z_n})\ \geq\ n.

The conjecture sits at the top of the known hierarchy of transcendence results. The Hermite–Lindemann theorem proves $e^{\alpha}$ transcendental for every nonzero algebraic $\alpha$ ; this is essentially the case $n=1$ of Schanuel's conjecture. The Lindemann–Weierstrass theorem strengthens this: exponentials of $\mathbb{Q}$ -linearly independent algebraic numbers are algebraically independent over the field of all algebraic numbers, so it already settles the algebraic-argument case of Schanuel's conjecture. The Gelfond–Schneider theorem, settling the transcendence of $\alpha^{\beta}$ for algebraic $\alpha\neq 0,1$ and irrational algebraic $\beta$ , is a consequence of the case $n=2$ of Schanuel's conjecture applied to logarithms of algebraic numbers; unlike the Lindemann–Weierstrass theorem, it concerns a single number rather than the algebraic independence of several exponential values. Beyond these, the strongest unconditional result toward the conjecture for the complex exponential is the six exponentials theorem: if $x_1,\dots,x_d$ are $\mathbb{Q}$ -linearly independent complex numbers and $y_1,\dots,y_{\ell}$ are $\mathbb{Q}$ -linearly independent complex numbers with $d\ell>d+\ell$ , then at least one of the $d\ell$ numbers $e^{x_i y_j}$ is transcendental; the four exponentials conjecture, asserting the same conclusion in the borderline case $d=\ell=2$ (where $d\ell=d+\ell$ and the theorem's hypothesis just fails), remains open and is itself a consequence of Schanuel's conjecture. The algebraic independence of $e$ and $\pi$ is open. Two neighboring programs supply unconditional results of Schanuel type in other settings. Ax's theorem (1971) proves the functional/differential analogue: in a differential field of characteristic zero (a field equipped with a derivation $f\mapsto f'$ , whose constant field is the subfield of elements with $f'=0$ ), functions $f_1,\dots,f_n$ that are $\mathbb{Q}$ -linearly independent modulo the constants — meaning no nontrivial rational linear combination of the $f_i$ is constant — together with their derivatives $f_1',\dots,f_n'$ generate an extension of transcendence degree at least $n$ over the constant field; this is the model for the now-proved Ax–Schanuel theorems for modular functions (special functions on moduli spaces of elliptic curves) and on Shimura varieties (higher-dimensional analogues of such moduli spaces). Zilber's program (2002–2005) constructs, via model theory, nonstandard 'Zilber fields' with a pseudo-exponentiation map satisfying Schanuel's property and strong structural conditions, and proposes proving that the complex exponential field $(\mathbb{C},+,\cdot,e^z)$ is such a field — a route that would prove the conjecture but whose required steps (notably exponential-algebraic closedness of $\mathbb{C}$ , the requirement that systems of exponential-polynomial equations over $\mathbb{C}$ satisfying the obvious necessary dimension constraints actually have solutions) remain open. The conjecture is known to imply the algebraic independence of $e$ and $\pi$ , the four exponentials conjecture, and algebraic independence statements for logarithms of algebraic numbers, and it is used as a standing hypothesis across number theory, model theory, and even dynamical-systems and digit-distribution questions.

Problem Statement

Prove or refute Schanuel's conjecture, stated in its full original generality with no restriction on $n$ or on the $z_i$ : for every positive integer $n$ and all complex numbers $z_1,\dots,z_n$ that are linearly independent over $\mathbb{Q}$ , the $2n$ numbers $z_1,\dots,z_n,e^{z_1},\dots,e^{z_n}$ contain at least $n$ numbers algebraically independent over $\mathbb{Q}$ ; equivalently,

\operatorname{trdeg}_{\mathbb{Q}}\ \mathbb{Q}\left(z_1,\dots,z_n,e^{z_1},\dots,e^{z_n}\right)\ \geq\ n

whenever $z_1,\dots,z_n$ are $\mathbb{Q}$ -linearly independent.

References

James Ax, 'On Schanuel's Conjectures', Annals of Mathematics (1971), DOI 10.2307/1970774, https://doi.org/10.2307/1970774
Michel Waldschmidt, 'Schanuel's Conjecture: algebraic independence of transcendental numbers', in 'Colloquium De Giorgi 2013 and 2014', Springer (2015), DOI 10.1007/978-88-7642-515-8_8, https://doi.org/10.1007/978-88-7642-515-8_8
B. Zilber, 'Exponential sums equations and the Schanuel conjecture', Journal of the London Mathematical Society (2002), DOI 10.1112/s0024610701002861, https://doi.org/10.1112/s0024610701002861
B. Zilber, 'Pseudo-exponentiation on algebraically closed fields of characteristic zero', Annals of Pure and Applied Logic (2005), DOI 10.1016/j.apal.2004.07.001, https://doi.org/10.1016/j.apal.2004.07.001
Paola D'Aquino, Angus Macintyre, Giuseppina Terzo, 'Schanuel Nullstellensatz for Zilber fields', Fundamenta Mathematicae 207 (2010), DOI 10.4064/fm207-2-2, https://doi.org/10.4064/fm207-2-2
Ngaiming Mok, Jonathan Pila, Jacob Tsimerman, 'Ax–Schanuel for Shimura varieties', Annals of Mathematics 189 (2019), DOI 10.4007/annals.2019.189.3.7, https://doi.org/10.4007/annals.2019.189.3.7
Michel Waldschmidt, 'Schanuel Property for Elliptic and Quasi-Elliptic Functions', arXiv:2504.14041 (2025), https://arxiv.org/abs/2504.14041
Cristiana Bertolin, Patrice Philippon, Biswajyoti Saha, Ekata Saha, 'Semi-abelian analogues of Schanuel Conjecture and applications', arXiv:2010.15170 (2020), https://arxiv.org/abs/2010.15170
Cristiana Bertolin, 'A conjecture in Schanuel style for 1-motives', arXiv:2509.08700 (2025), https://arxiv.org/abs/2509.08700
Wayne M. Lawton, 'Joint distribution of leftmost digits in positional notation and Schanuels's conjecture', arXiv:2603.03110 (2026), https://arxiv.org/abs/2603.03110