Thanks a lot, Franklin and Gareth. I suspected that conditions (*) and (**) together would result in algebraic coordinates, mainlly because I'd expect to have heard of these numbers if they defined a larger class. The theorem Gareth proves seems very basic, so I wonder if it has a name and attribution. Also, given such an integer-polynomial function P: C^n -> C^n such that its Jacobian at each of its roots Z in C^n is non-singular: QUESTION: What can be said about the size of the (necessarily) 0-dimensional set of such roots? (Where of course a root of P means a Z in C^n where P(Z) is the origin in C^n.) --Dan