# What does $\mathbb{Z}[X]$ for a polynom mean?

I have a proof saying that a Polynom $p \in \mathbb{Z}[X_1,...,X_m]$

I'm a bit confused of this notation because neither X nor the m is explains somewhere.

Does somebody of you know the notation?

• – Emilio Novati Apr 17 '16 at 13:44
• $X$ is just a variable and $X_1$ through $X_m$ just means that $p$ is a polynomial with $m$ variables. For example, $p(X_1, X_2, ..., X_m)=X_1^2X_2^2+2X_2^2X_3^2+3X_3^2X_4^2+[...]+(m-1)X_{m-1}^2X_m^2$ is one possibility. If it says $\Bbb{Z}[X_1, X_2, X_3, ..., X_m]$, that means all of the coefficients have to be integers. – Noble Mushtak Apr 17 '16 at 13:44
• @NobleMushtak Correction: $X$ is not the variable. $X_1, \ldots, X_m$ are the variables. – Anon Apr 17 '16 at 13:50
• @McFry: Correction to correction: the $X_i$ are not variables, but indeterminates. A polynomial is different from a polynomial function. – Bernard Apr 17 '16 at 14:09

If it helps, think of $\mathbb{Z}[X_1,X_2,X_3]$ as $\mathbb{Z}[x,y,z]$ and then your favorite polynomials (in three variables) from multivariable calculus are in this ring. For instance, for $z^2=x^2+y^2$, you have $z^2-x^2-y^2 \in \mathbb{Z}[x,y,z]$. But for an arbitrary polynomial in $m$ variables it is better to write $X_1,...,X_m$ than choosing $m$ symbols for each variable. Once the notation has been established, one writes $X$ in place of $X_1,...,X_m$. So $\mathbb{Z}[X_1,...,X_m]=\mathbb{Z}[X]$ and $f(X_1,...,X_m)\in \mathbb{Z}[X_1,...,X_m]$ is simply written as $f(X) \in \mathbb{Z}[X]$.
This means that $p$ is a polynomial with integer coefficients in $m$ variables. An example would be $$p( X_1 , X_2 , \ldots , X_m) = X_1 + X_2 + \cdots + X_m.$$