Schanuel's Conjecture

Let , ...,  be linearly independent over the rationals , then

has transcendence degree at least  over . Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that  and  are algebraically independent. Macintyre (1991) proved that the truth of Schanuel's conjecture also guarantees that there are no unexpected exponential-algebraic relations on the integers  (Marker 1996).

At present, a proof of Schanuel's conjecture seems out of reach (Chow 1999).

REFERENCES:

Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440-448, 1999.

Chudnovsky, G. V. "On the Way to Schanuel's Conjecture." Ch. 3 in Contributions to the Theory of Transcendental Numbers. Providence, RI: Amer. Math. Soc., pp. 145-176, 1984.

Lin, F.-C. "Schanuel's Conjecture Implies Ritt's Conjecture." Chinese J. Math. 11, 41-50, 1983.

Macintyre, A. "Schanuel's Conjecture and Free Exponential Rings." Ann. Pure Appl. Logic 51, 241-246, 1991.

Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753-759, 1996.