World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0002031353
Reproduction Date:

Title: Schanuel  
Author: World Heritage Encyclopedia
Language: English
Subject: E (mathematical constant), Exponential field, Euler's identity, James Ax, Exponential function
Publisher: World Heritage Encyclopedia


In mathematics, specifically transcendence theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.


The conjecture is as follows:

Given any n complex numbers z1,...,zn that are linearly independent over the rational numbers Q, the extension field Q(z1,...,zn, exp(z1),...,exp(zn)) has transcendence degree of at least n over Q.

The conjecture can be found in Lang (1966).[1]


The conjecture, if proven, would generalize most known results in transcendental number theory. The special case where the numbers z1,...,zn are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(z1),...,exp(zn) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem.

The Gelfond–Schneider theorem follows from this strengthened version of Baker's theorem, as does the currently unproven four exponentials conjecture.

Schanuel's conjecture, if proved, would also settle the algebraic nature of numbers such as e + π and ee, and prove that e and π are algebraically independent simply by setting z1 = 1 and z2 = πi, and using Euler's identity.

Euler's identity states that eπi + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, π, and i over the complex numbers.[2]

Although ostensibly a problem in number theory, the conjecture has implications in model theory as well. Angus Macintyre and Alex Wilkie, for example, proved that the theory of the real field with exponentiation, Rexp, is decidable provided Schanuel's conjecture is true.[3] In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem.

Related conjectures and results

The converse Schanuel conjecture[4] is the following statement:

Suppose F is a countable field with characteristic 0, and e : FF is a homomorphism from the additive group (F,+) to the multiplicative group (F,·) whose kernel is cyclic. Suppose further that for any n elements x1,...,xn of F which are linearly independent over Q, the extension field Q(x1,...,xn,e(x1),...,e(xn)) has transcendence degree at least n over Q. Then there exists a field homomorphism h : FC such that h(e(x))=exp(h(x)) for all x in F.

A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971.[5] It states:

Given any n formal power series f1,...,fn in tC which are linearly independent over Q, then the field extension C(t,f1,...,fn,exp(f1),...,exp(fn)) has transcendence degree at least n over C(t).

As stated above, the decidability of Rexp follows from the real version of Schanuel's conjecture which is as follows:[6]

Suppose x1,...,xn are real numbers and the transcendence degree of the field Q(x1,...,xn, exp(x1),...,exp(xn)) is strictly less than n, then there are integers m1,...,mn, not all zero, such that m1x1 +...+ mnxn = 0.

A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers mi. The uniform real version of the conjecture is equivalent to the standard real version.[6] 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 Rexp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomials; this is, non-obviously, a consequence of Schanuel's conjecture for the reals.[3]

It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives. There Grothendieck's period conjecture for an abelian variety A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford–Tate group, and what is known by work of Pierre 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.[7]

Zilber's pseudo-exponentiation

While a proof of Schanuel's conjecture with number theoretic tools seems a long way off,[8] connections with model theory have prompted a surge of research on the conjecture.

In 2004, Boris Zilber systematically constructs exponential fields Kexp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality.[9] He axiomatises these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves 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 shows that this conjecture holds iff both Schanuel's conjecture and another unproven condition on the complex exponentiation field, which Zilber calls exponential-algebraic closedness, hold.[10]


  1. ^  
  2. ^ Terzo, Giuseppina (2008). "Some consequences of Schanuel's conjecture in exponential rings". Communications in Algebra 36 (3): 1171–1189.  
  3. ^ a b Macintyre, A. & Wilkie, A. J. (1996). "On the decidability of the real exponential field". In Odifreddi, Piergiorgio. Kreiseliana: About and Around Georg Kreisel. Wellesley: Peters. pp. 441–467.  
  4. ^ Scott W. Williams, Million Bucks Problems
  5. ^ Ax, James (1971). "On Schanuel's conjectures".  
  6. ^ a b Kirby, Jonathan & Zilber, Boris (2006). "The uniform Schanuel conjecture over the real numbers". Bull. London Math. Soc. 38: 568–570.  
  7. ^ Bertolin, Cristiana (2002). "Périodes de 1-motifs et transcendance". Journal of Number Theory 97 (2): 204–221.  
  8. ^ Waldschmidt, Michel (2000). Diophantine approximation on linear algebraic groups. Berlin:  
  9. ^ Zilber, Boris (2004). "Pseudo-exponentiation on algebraically closed fields of characteristic zero".  
  10. ^ Zilber, Boris (2002). "Exponential sums equations and the Schanuel conjecture". J. London Math. Soc. 65 (2): 27–44.  

External links

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.