TITLE: Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
QUESTION [7 upvotes]: Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the definition of $f(x)$? That is, can we find a polynomial with one variable satisfying these properties?
Actually, it would be better if there is a general method for finding polynomials whose values are determined like this one. For $\mathbb Z_2$, there is a method called butterfly algorithm. Using this, one can find the algebraic normal form of a function using the values (truth table of that function).
Thank you.

REPLY [3 votes]: If $f(X)\in \mathbb{Z}_9[X]$ has $f(6)=0$, then $f(X)=(X-6)g(X)$ for some $g(X)$ so $f(0)=3g(0)\neq 1.$