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Henson and Rubel's theorem for Zilber's pseudoexponentiation

  • Ahuva C. Shkop (a1)

Abstract

In 1984, Henson and Rubel [2] proved the following theorem: If p(x1,…, xn) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x1,…, xn) = eg(x1,…, xn) where g(x1,…, xn) is some exponential polynomial over C. In this paper, I will prove the analog of this theorem for Zilber's Pseudoexponential fields directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel's conjecture.

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References

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[1] D'aquino, P., Macintyre, A., and Terzo, G., Schanuel Nullstellensatz for Zilber fields, Fundamenta Mathematical, vol. 207 (2010), pp. 123143.
[2] Henson, C. W. and Rubel, L. A., Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions, Transactions of the American Mathematical Society, vol. 282 (1984), no. 1, pp. 132.
[3] Macintyre, Angus, Schanuel's conjecture and free exponential rings. Annals of Pure and Applied Logic, vol. 51 (1991), no. 3, pp. 241246.
[4] Marker, David, Remarks on Zilber's pseudoexponentiation, this Journal, vol. 71 (2006), no. 3, pp. 791798.
[5] Shafarevich, Igor, Basic algebraic geometry, Springer-Verlag Telos, 1995.
[6] Van Den Dries, Lou, Exponential rings, exponential polynomials and exponential functions. Pacific Journal of Mathematics, vol. 113 (1984), no. 1, pp. 5166.
[7] Zilber, Boris, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2005), no. 1, pp. 6795.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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