# Are polynomials over a finite field generally functions from that field or…?

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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$X^3 + X^2 - X + 1$ is an element in the ring $\mathbb{Z}_{3}[X].$ The function induced by evaluating the polynomial expression at $X \in \mathbb{Z}_3$ is $g(X).$ You can consider that as an evaluation map, or a function, $g : \mathbb{Z}_{3} \to \mathbb{Z}_{3}.$ And I guess it's total since $\mathbb{Z}_{3}$ is closed under all the operations in the expression for $g(X),$ i.e., closed under $+, -, *$. –  user2468 Mar 18 '12 at 6:21
Re: your Edit. It's difficult to infer what you are asking without any definition of what it means for polynomials to be of the same "form" or be "like". Perhaps it would help to give some examples illustrating what you have in mind. –  Bill Dubuque Mar 18 '12 at 15:10

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 $\mathbb{F}_3 \times \mathbb{F}_{27}$.