Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

Abstract

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.

Key Words:

• Exponential algebra
• Schanuel's conjecture
• Shapiro's conjecture

2000 Mathematics Subject Classification:

• 03C60
• 11U09

ACKNOWLEDGMENTS

This work was done as part of my thesis and I would like to thank my advisor, David Marker, for all of his guidance and support. I would also like to thank Alf Dolich for many helpful comments on preliminary versions of this article. I would also like to thank Lou van den Dries and Ward Henson for pointing out this last note to me.

Notes

Communicated by D. Macpherson.