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Schanuel's Conjecture

Let lambda_1, ..., lambda_n in C be linearly independent over the rationals Q, then

Q(lambda_1,...,lambda_n,e^(lambda_1),...,e^(lambda_n))

has transcendence degree at least n over Q. Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that e and pi 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 Z (Marker 1996).

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

See also

Algebraically Independent, Constant Problem, Gelfond's Theorem, Lindemann-Weierstrass Theorem, Richardson's Theorem, Uniformity Conjecture

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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.

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Schanuel's Conjecture

Cite this as:

Weisstein, Eric W. "Schanuel's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchanuelsConjecture.html

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