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On the ring of differentially-algebraic entire functions

Published online by Cambridge University Press:  12 March 2014

Lee A. Rubel
Lee A. Rubel*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, E-mail: rubel@symcom.math.uiuc.edu
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Extract

Let be the ring of all entire functions of one complex variable, and let DA be the subring of those entire functions that are differentially algebraic (DA); that is, they satisfy a nontrivial algebraic differential equation.

where P is a non-identically-zero polynomial in its n + 2 variables. It seems not to be known whether DA is elementarily equivalent to . This would mean that DA and have exactly the same true statements about them, in the first-order language of rings. (Roughly speaking, a sentence about a ring R is first-order if it has finite length and quantifies only over elements (i.e., not subsets or functions or relations) of R.) It follows from [NAN] that DA and are not isomorphic as rings, but this does not answer the question of elementary equivalence.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 449 - 451
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[NAN]Nandakumar, N. R., Ring homomorphisms on algebras of analytic functions, preprint, 1990.Google Scholar
[BHR]Becker, J., Henson, C. W., and Rubel, L. A., First-order conformai invariants, Annals of Mathematics, ser. 2, vol. 112 (1980), pp. 123–178.CrossRefGoogle Scholar