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Suppose we are considering polynomials in $\mathbb{Z}_n [x]$. Is there a way to decide if two polynomials determine the same function from $\mathbb{Z}_n$ to itself without having to plug in all numbers in $\mathbb{Z}_n$.

For example in $\mathbb{Z}_3 [x]$, I know that the polynomials $x^4 +x$ and $x^2 + x$ define the same function from $\mathbb{Z}_3$ to itself. Is there any way to prove this without plugging in all values in $\mathbb{Z}_3$.

tmbbdil
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  • Hint: Well, why not subtract them and see if it's constant zero? That'd be way easier... – Trebor Feb 18 '22 at 04:37
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    The cases where $n$ is prime are easier to describe than for composite $n$. – hardmath Feb 18 '22 at 04:38
  • If you subtract them and are able to factor it, check to see if all members of $\mathbb Z_3$ are zeros of that difference. In your case, you have $x^4-x^2=x^2(x-1)(x+1)=(x-0)^2(x-1)(x-2)$ and all elements of $\mathbb Z_3$ are zeros. – David P Feb 18 '22 at 04:45
  • Note that for $p$ prime any two polynomials in degree at most $p-1$ are distinct as functions and any polynomial in degree $p$ or higher can be reduced to one like that by using $x^p-x=0$ so one can do the reduction and check coefficients eg for $p=3$ obviously $x^4$ reduces to $x^2$ so the op claim is immediate – Conrad Feb 18 '22 at 04:53
  • Closely related to https://math.stackexchange.com/questions/3387540/polynomials-that-induce-the-zero-function-mod-n – lhf Feb 18 '22 at 09:36
  • This link pretty much shows why this question is not trivial: https://math.stackexchange.com/questions/4070051/how-many-distinct-polynomials-are-there-mod-n/4070413#4070413 – Chris Sanders Feb 18 '22 at 05:21
  • See this answer of mine; as far as I can tell the question is the same? – ShreevatsaR Feb 19 '22 at 04:26

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