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Take $g(X) = X^3 + X^2 - X + 1$ and let it be over $\mathbb{Z}_3$, what is usually the domain of $g$?

Grazie.

Edit: What I want to do is come up with an intelligent way t think of all the polynomials of a certain form, say $VW + XY + Z^2$ (this time in $(\mathbb{Z}_2$ or $\mathbb{Z})[V,W,X,Y,Z]$), but generalize that "form" slightly by looking at other polynomials that are "like" it. So first I'm looking at the rings of polynomials. To restrict the size of the ring I'm thinking a ring mod $X^q - 1$ (or whatever a corresponding multivariate modulus would be). So all the possible poly expressions are listable on a computer in decent time.

I figure a smart way might involve looking at factors of these expressions if I'm dealing with rational poly expressions: $P(V,\ldots,Z) / Q(V,\ldots,Z)$. I don't want to observe one possibility more than once.

Other things might make equal polys in this "set of polys that I'm interested in" like rearrangement of terms, and possibly permutations of the variables.

Once I have all this down. Then what we have is a likely set of expressions in which we might find a Somos Sequence

This is all obvious, but would be good practice for me (a novice programmer) to do. But if anyone else has already done this, I shouldn't bother. So if anyone knows, please post a link.

Thanks.

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    You may be very interested in ["The algebraic structure of rational discrete dynamical systems"](https://arxiv.org/abs/1501.06384) arXiv.org:1501.06384 which studies several examples of rational polynomial expressions.2018-07-19

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Polynomials are polynomials, not functions. Taken very literally, it doesn't make sense to ask for their domain.

In common usage, a polynomial over $\mathbb{F}_3$ can be evaluated in any algebra over $\mathbb{F}_3$. e.g. people will write $g(a)$ when $a \in \mathbb{F}_3$, when $a \in \mathbb{F}_9$, when $a \in \overline{\mathbb{F}_3}$, or even when $a$ is a 17 by 17 matrix whose entries are bivariate polynomials with coefficients in $\mathbb{F}_3 \times \mathbb{F}_{27}$.

(and there are even other circumstances aside from this one would write such things!)

If you ever see someone talk about the domain of a polynomial (and thus intending to view it as a function), your best bet is to try and infer it from context. As others mentioned, the coefficient ring is often a reasonable guess.