Is a multilinear polynomial of variables $x_1, \dots, x_n$ over a ring defined as a monomial $c \prod_{i=1}^n x_i$, where $c$ is a constant from the ring?
Equivalently, is a multilinear polynomial function of variables $x_1, \dots, x_n$ over a ring same as a multilinear mapping of $x_1, \dots, x_n$?
My confusion comes from Wikipedia
In algebra, a multilinear polynomial is a polynomial that is linear in
each of its variables. In other words, no variable occurs to a power
of 2 or higher; or alternatively, each monomial is a constant times a
product of distinct variables. … The degree of a multilinear
polynomial is the maximum number of distinct variables occurring in
any monomial.
It seems to suggest a multilinear polynomial can have more than one monomial terms. Thanks!
Best Answer
According to this definition, as Arturo said, a multilinear polynomial in $k[x_1,x_2,\ldots,x_n]$ is not the same as a polynomial that induces a multilinear mapping on $k^n$, the latter being only $c\prod\limits_{k=1}^n x_k$ as you indicated. There aren't many interesting multilinear maps from $k^n\to k$ when $k$ is a field, but the multilinear polynomials according to this definition include any polynomial such that each monomial is squarefree. This means that fixing $n-1$ of the variables induces an affine map $k\to k$, i.e. a map of the form $x\mapsto ax+b$.
On the other hand, if $R$ is a noncommutative ring, then there may be more interesting multilinear maps $R^n\to R$, and consequently reason for restricting the definition of multilinear polynomial to mean one that induces a multilinear map. But in this case the polynomials should be in noncommuting variables. E.g., if $R$ is a $k$-algebra, then elements of the "noncommutative polynomial ring" (i.e., free algebra) $k\langle x_1,x_2,\ldots,x_n\rangle$ induce maps $R^n\to R$, and include nonmonomial multilinear polynomials (in the restricted sense) like $x_1x_2\cdots x_n + x_n x_{n-1}\cdots x_1$. Such (noncommutative) multilinear polynomials are important in the theory of polynomial identity rings.