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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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Multilinear polynomials can certainly have more than one monomial. For example, $xy+3xyz + 2yz + 6x + 8y$ is a multilinear polynomial. The key is that each monomial should be squarefree. –  Arturo Magidin Jan 16 '12 at 20:31
@ArturoMagidin: Thanks! But $(x_1+x_2)y+3(x_1+x_2)yz+2yz+6(x_1+x_2)+8y$ and $(x_1y+3x_1yz+2yz+6x_1+8y) + (x_2y+3x_2yz+2yz+6x_2+8y)$ are not equal, which means the polynomial is not linear in $x$. This comes to the question of how you definite a multilinear polynomial. –  Tim Jan 16 '12 at 20:36
It's an affine transformation in $x$ (the terms $2yz$ and $8y$ are constant, which is what "throws off" pure linearity). Just like the function $f(x) = 3x+1$ is not a "linear transformation" in $x$, but it is nonetheless called a "linear function". You are running into two different uses of "linear" here. Here, "linear" refers to "degree 1 polynomial", not to "additive and homogeneous". –  Arturo Magidin Jan 16 '12 at 20:38
@Arturo: Thanks! Just to be sure: When talking about polynomials, "linear" function means affine mapping between affine spaces? When talking about multilinear mapping, "linear in each vector" means linear mapping between vector spaces? When talking about linear combination, "linear" also means linear mapping between vector spaces? –  Tim Jan 16 '12 at 20:45
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". –  Arturo Magidin Jan 16 '12 at 20:46
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up vote 2 down vote accepted

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.

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