This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate? Excessive Violence Sexual Content Political / Social
Email Address:
Article Id: WHEBN0002031353 Reproduction Date:
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:
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 z_{1},...,z_{n} are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(z_{1}),...,exp(z_{n}) 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 e^{e}, and prove that e and π are algebraically independent simply by setting z_{1} = 1 and z_{2} = πi, and using Euler's identity.
Euler's identity states that e^{πi} + 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, R_{exp}, 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.
The converse Schanuel conjecture^{[4]} is the following statement:
A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971.^{[5]} It states:
As stated above, the decidability of R_{exp} follows from the real version of Schanuel's conjecture which is as follows:^{[6]}
A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers m_{i}. 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 R_{exp}. 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]}
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 K_{exp} 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]}
Logic, Set theory, Statistics, Number theory, Mathematical logic
Pi, Leonhard Euler, Calculus, Irrational number, Exponential function
E (mathematical constant), Derivative, Physics, Chemistry, Economics
Pi, E (mathematical constant), Electromagnetism, Euler's formula, Leonhard Euler
Mathematics, Number theory, E (mathematical constant), Schanuel's conjecture, Model theory
University of California, San Diego, Algebra, Number theory, University of California, Berkeley, Cornell University