Inmathematics, specificallytranscendental number theory,Schanuel's conjecture is aconjecture about thetranscendence degree of certainfield extensions of therational numbers${\displaystyle \mathbb{Q}}$, which would establish thetranscendence of a large class ofnumbers, for which this is currentlyunknown. It is due toStephen Schanuel and was published bySerge Lang in 1966.^[1]

Schanuel's conjecture can be given as follows:^[1][2]

Schanuel's conjecture, if proven, would generalize most known results intranscendental number theory and establish a large class of numbers transcendental. Special cases of Schanuel's conjecture include:

Considering Schanuel's conjecture for only${\displaystyle n=1}$ gives that for nonzero complex numbers${\displaystyle z}$, at least one of the numbers${\displaystyle z}$ and${\displaystyle e^z}$ must be transcendental. This was proved byFerdinand von Lindemann in 1882.^[3]

If the numbers${\displaystyle z_1,...,z_n}$ are taken to be allalgebraic andlinearly independent over${\displaystyle \mathbb{Q}}$ then the${\displaystyle e^{z_1},...,e^{z_n}}$result to be transcendental andalgebraically independent over${\displaystyle \mathbb{Q}}$. The first proof for this more general result was given byCarl Weierstrass in 1885.^[4]

This so-calledLindemann–Weierstrass theorem implies the transcendence of the numberse andπ. It also follows that for algebraic numbers${\displaystyle \alpha }$ not equal to0 or1, both${\displaystyle e^{\alpha }}$ and${\displaystyle \ln (\alpha )}$ are transcendental. It further gives the transcendence of thetrigonometric functions at nonzero algebraic values.

Another special case was proved byAlan Baker in 1966: If complex numbers${\displaystyle {\lambda }_1,...,{\lambda }_n}$ are chosen to be linearly independent over the rational numbers${\displaystyle \mathbb{Q}}$ such that${\displaystyle e^{{\lambda }_1},...,e^{{\lambda }_n}}$ are algebraic, then${\displaystyle {\lambda }_1,...,{\lambda }_n}$ are also linearly independent over the algebraic numbers${\displaystyle \overline{\mathbb{Q}}}$.

Schanuel's conjecture would strengthen this result, implying that${\displaystyle {\lambda }_1,...,{\lambda }_n}$ would also be algebraically independent over${\displaystyle \mathbb{Q}}$ (and equivalently over${\displaystyle \overline{\mathbb{Q}}}$).^[2]

In 1934 it was proved byAleksander Gelfond andTheodor Schneider that if${\displaystyle \alpha }$ and${\displaystyle \beta }$ are two algebraic complex numbers with${\displaystyle \alpha \notin \{0,1\}}$ and${\displaystyle \beta \notin \mathbb{Q}}$, then${\displaystyle {\alpha }^{\beta }}$ is transcendental.

This establishes the transcendence of numbers likeHilbert's constant${\displaystyle 2^{\sqrt{2}}}$ andGelfond's constant${\displaystyle e^{\pi }}$.^[5]

TheGelfond–Schneider theorem follows from Schanuel's conjecture by setting${\displaystyle n=3}$ and${\displaystyle z_1=\beta ,z_2=\ln \alpha ,z_3=\beta \ln \alpha }$. It also would follow from the strengthened version of Baker's theorem above.

The currently unprovenfour exponentials conjecture would also follow from Schanuel's conjecture: If${\displaystyle z_1,z_2}$ and${\displaystyle w_1,w_2}$ are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers istranscendental:

$${\displaystyle e^{z_1{w}_1},e^{z_1{w}_2},e^{z_2{w}_1},e^{z_2{w}_2}.}$$

The four exponential conjecture would imply that for any irrational number${\displaystyle t}$, at least one of the numbers${\displaystyle 2^t}$ and${\displaystyle 3^t}$ is transcendental. It also implies that if${\displaystyle t}$ is a positive real number such that both${\displaystyle 2^t}$ and${\displaystyle 3^t}$ are integers, then${\displaystyle t}$ itself must be an integer.^[2] The relatedsix exponentials theorem has been proven.

Schanuel's conjecture, if proved, would also establish many nontrivial combinations ofe,π, algebraic numbers andelementary functions to be transcendental:^[2][6][7]

$${\displaystyle e+\pi ,e\pi ,e^{{\pi }^2},e^e,{\pi }^e,{\pi }^{\sqrt{2}},{\pi }^{\pi },{\pi }^{{\pi }^{\pi }},\log \pi ,\log \log 2,\sin e,...}$$

In particular it would follow thate andπ are algebraically independent simply by setting${\displaystyle z_1=1}$ and${\displaystyle z_2=i\pi }$.

Euler's identity states that${\displaystyle e^{i\pi }+1=0}$. If Schanuel's conjecture is true then this is, in some precise sense involvingexponential rings, theonly non-trivial relation betweene,π, andi over the complex numbers.^[8]

Theconverse Schanuel conjecture^[9] is the following statement:

SupposeF is acountablefield withcharacteristic 0, ande :F →F is ahomomorphism from the additive group (F,+) to the multiplicative group (F,·) whosekernel iscyclic. Suppose further that for anyn elementsx₁,...,x_n ofF which are linearly independent over${\displaystyle \mathbb{Q}}$, the extension field${\displaystyle \mathbb{Q}}$(x₁,...,x_n,e(x₁),...,e(x_n)) has transcendence degree at leastn over${\displaystyle \mathbb{Q}}$. Then there exists a field homomorphismh :F →${\displaystyle \mathbb{C}}$ such thath(e(x)) = exp(h(x)) for allx inF.

A version of Schanuel's conjecture forformal power series, also by Schanuel, was proven byJames Ax in 1971.^[10] It states:

Given anyn formalpower seriesf₁,...,f_n int${\displaystyle \mathbb{C}}$[[t]] which are linearly independent over${\displaystyle \mathbb{Q}}$, then the field extension${\displaystyle \mathbb{C}}$(t,f₁,...,f_n,exp(f₁),...,exp(f_n)) has transcendence degree at leastn over${\displaystyle \mathbb{C}}$(t).

Although ostensibly a problem in number theory, Schanuel's conjecture has implications inmodel theory as well.Angus Macintyre andAlex Wilkie, for example, proved that the theory of the real field with exponentiation,${\displaystyle \mathbb{R}}$_exp, isdecidable provided Schanuel's conjecture is true.^[11] In fact, to prove this result, they only needed the real version of the conjecture, which is as follows:^[12]

Supposex₁,...,x_n arereal numbers and the transcendence degree of the field${\displaystyle \mathbb{Q}}$(x₁,...,x_n,exp(x₁),...,exp(x_n)) is strictly less thann, then there are integersm₁,...,m_n, not all zero, such thatm₁x₁ +...+ m_nx_n = 0.

This would be a positive solution toTarski's exponential function problem.

A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integersm_i. The uniform real version of the conjecture is equivalent to the standard real version.^[12] Macintyre and Wilkie showed that a consequence of Schanuel's conjecture, which they dubbed the Weak Schanuel's conjecture, was equivalent to the decidability of${\displaystyle \mathbb{R}}$_exp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems ofexponential polynomials; this is, non-obviously, a consequence of Schanuel's conjecture for the reals.^[11]

It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory ofmotives. In this settingGrothendieck's period conjecture for anabelian varietyA states that the transcendence degree of itsperiod matrix is the same as the dimension of the associatedMumford–Tate group, and what is known by work ofPierre Deligne is that the dimension is an upper bound for the transcendence degree. Bertolin has shown how a generalised period conjecture includes Schanuel's conjecture.^[13]

While a proof of Schanuel's conjecture seems a long way off, as reviewed by Michel Waldschmidt in the year 2000,^[14] connections with model theory have prompted a surge of research on the conjecture.

In 2004,Boris Zilber systematically constructedexponential fieldsK_exp that are algebraically closed and of characteristic zero, and such that one of these fields exists for eachuncountablecardinality.^[15] He axiomatised these fields and, usingHrushovski's construction and techniques inspired by work of Shelah oncategoricity ininfinitary logics, proved that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber showed that this conjecture holds if and only if both Schanuel's conjecture and theExponential-Algebraic Closedness conjecture hold.^[16] As this construction can also give models with counterexamples of Schanuel's conjecture, this method cannot prove Schanuel's conjecture.^[17]

• Four exponentials conjecture
• Algebraic independence
• List of unsolved problems in mathematics
• Existential Closedness conjecture
• Zilber-Pink conjecture
• Pregeometry

• Weierstrass, K. (1885),"Zu Lindemann's Abhandlung. "Über die Ludolph'sche Zahl".",Sitzungsberichte der Königlich Preussischen Akademie der Wissen-schaften zu Berlin,5:1067–1085

• Weisstein, Eric W."Schanuel's Conjecture".MathWorld.

• Conjectures
• Unsolved problems in number theory
• Exponentials
• Transcendental numbers

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