What Sort of Object is zl

When one first looks at a polynomial the expectation is that it is a function, a map from a domain to a range, and defining the domain is an important part of the semantics of the function. 

It is possible to use the idea of evaluating the polynomial at a real or complex value as an aid to proving that one polynomial is a factor of another, by showing that all the roots of the first are also roots of the second. In fact in the Laplace transform the domain is definitely that of complex numbers, and [CDM91] uses this interpretation very fluently and to good effect. 

However, there are four other interpretations -1 A purely algebraic symbol 

This interpretation says that the only reason we are playing with these polynomials is to determine coefficients of other polynomials. Formal manipulations, in which the nature of the variable plays no part, and so we do not need to define anything about it, beyond the property that it can be raised to positive or negative powers. 

This is strictly correct in this context, but anybody who trumpets it too loud had better find other ways of proving divisibility of one polynomial by another than that just mentioned. 

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