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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!

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    When talking about polynomials, "constant" means "degree 0", "linear" means "degree 1", "quadratic" means "degree 2", "cubic" means "degree 3", etc. When talking about multilinear polynomials, "linear in each variable" means "degree 1 in each variable".2012-01-16

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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.