Schanuel's conjecture
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In mathematics, specifically transcendence theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of a certain field extension of the rational numbers.
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[edit] Statement
The conjecture is as follows:
- Given any n complex numbers z1,...,zn which 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]. No proof is known.
[edit] Consequences
The conjecture, if proven, would subsume 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.
[edit] Related conjectures and results
The converse Schanuel conjecture[4] is the following statement:
- Suppose F is a countable field with characteristic 0, and e : F → F 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 : F → C 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[[t]] 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, a fact that is not an immediately obvious consequence of Schnauel's conjecture.[3]
[edit] Zilber's pseudo-exponentiation
While a proof of Schanuel's conjecture with number theoretic tools seems a long way off[7], an approach via model theory has prompted a surge of research on the conjecture.
In 2004, Boris Zilber systematically constructed exponential fields Kexp that were algebraically closed and of characteristic zero, and such that one of these fields existed for each uncountable cardinality.[8] He axiomatised these fields and, using Hrushovski's construction, he proved that for each uncountable cardinal this theory of "pseudo-exponentiation" is satisfiable, and categorical for a particular field, this field he called the canonical one for that cardinality. The canonical field for the cardinality of the continuum satisfies Schanuel's conjecture, and so if in this case Kexp = Cexp then Schanuel's conjecture is true. Unfortunately one of the criteria for this model to be the complex numbers with exponentiation is that Schanuel's conjecture is true, as well as another unproven property of the complex numbers with exponentiation, which Zilber calls exponential-algebraic closedness.[9]
[edit] References
- ^ Serge Lang, Introduction to Transcendental Numbers, Addison-Wesley, 1966. Pages 30-31
- ^ Giuseppina Terzo, Some consequences of Schanuel's conjecture in exponential rings, Communications in Algebra, 36:3, pages 1171–1189.
- ^ a b A. Macintyre and A. J. Wilkie, On the decidability of the real exponential field, Kreiseliana, 1996.
- ^ Scott W. Williams, Million Bucks Problems
- ^ James Ax, On Schanuel's conjectures, Annals of Mathematics(2) 93, 1971, pages 252-268.
- ^ a b Jonathan Kirby and Boris Zilber, The uniform Schanuel conjecture over the real numbers, Bull. London Math. Soc. 38 (2006), pages 568–570.
- ^ Michel Waldschmidt, Diophantine approximation on linear algebraic groups, Springer, 2000.
- ^ Boris Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, (132), 2004, 1, pages 67–95.
- ^ Boris Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), pages 27–44.
