It seems obvious that if two polynomials over integers are equivalent, then the same two polynomials over naturals must also be equivalent, by seeing the polynomials as syntactic expressions. But I can't see how to show this rigorously...
Perhaps, if the two polynomials are equivalent, there's a sequence of algebraic manipulations transforming one into the other. The same sequence will do the same thing for the two polynomials over naturals - provided all the transformations are defined (so no subtractions resulting in negative values).
This came up in the context of using the Schwartz Zippel lemma to show polynomials are equal (by showing their subtraction is always zero). My trouble is showing why this still works when subtraction isn't always defined, as for positive integers, i.e. natural numbers. The issue arose from: Is there an efficient algorithm for expression equivalence?