Chris Landauer writes: << i'm thinking of a (yet-to-be-characterized) ring of functions that includes the polynomials and also a few other basic functions (e.g., real polynomials with the exponential), and that is also closed under composition (as polynomial rings are, suitably interpreted)

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I don't know much at all about commutative algebra or algebraic geometry. But a natural ring containing the ring of real polynomials over real variables x_1, . . ., x_n is the ring of real analytic functions on R^n (power series in x_1, . . ., x_n convergent on all of R^n). (In fact, if you want your ring to be closed under pointwise limits, then the real analytic functions form the smallest such ring -- and they clearly contain the polynomials and exponentials, and are closed under composition.) There does appear to be a nullstellensatz in this case: see e.g. << Adkins,William A.; Leahy, J.V. A global real analytic nullstellensatz. DukeMath. J. 43 (1976), no. 1, 81–86.

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--Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele