In algebra, a multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of ${\displaystyle 2}$ or higher; that is, each monomial is a constant times a product of distinct variables. For example ${\displaystyle f(x,y,z)=3xy+2.5y-7z}$ is a multilinear polynomial of degree ${\displaystyle 2}$ (because of the monomial ${\displaystyle 3xy}$) whereas ${\displaystyle f(x,y,z)=x^{2}+4y}$ is not. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.

Multilinear polynomials can be understood as a multilinear map (specifically, a multilinear form) applied to the vectors [1 x], [1 y], etc. The general form can be written as a tensor contraction:$${\displaystyle f(x)=\sum _{i_{1}=0}^{1}\sum _{i_{2}=0}^{1}\cdots \sum _{i_{n}=0}^{1}a_{i_{1}i_{2}\cdots i_{n}}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}}$$

For example, in two variables:$${\displaystyle f(x,y)=\sum _{i=0}^{1}\sum _{j=0}^{1}a_{ij}x^{i}y^{j}=a_{00}+a_{10}x+a_{01}y+a_{11}xy={\begin{pmatrix}1&x\end{pmatrix}}{\begin{pmatrix}a_{00}&a_{01}\\a_{10}&a_{11}\end{pmatrix}}{\begin{pmatrix}1\\y\end{pmatrix}}}$$

A multilinear polynomial ${\displaystyle f}$ is linear (affine) when varying only one variable, ${\displaystyle x_{k}}$:$${\displaystyle f(x_{1},x_{2},...,x_{k},...,x_{n})=ax_{k}+b}$$where ${\displaystyle a}$ and ${\displaystyle b}$ do not depend on ${\displaystyle x_{k}}$. Note that ${\displaystyle b}$ is generally not zero, so ${\displaystyle f}$ is linear in the "shaped like a line" sense, but not in the "directly proportional" sense of a multilinear map.

All repeated second partial derivatives are zero:$${\displaystyle {\frac {\partial ^{2}f}{\partial x_{k}^{2}}}=0}$$In other words, its Hessian matrix is a symmetric hollow matrix.

In particular, the Laplacian ${\displaystyle \nabla ^{2}f=0}$, so ${\displaystyle f}$ is a harmonic function. This implies ${\displaystyle f}$ has maxima and minima only on the boundary of the domain.

More generally, every restriction of ${\displaystyle f}$ to a subset of its coordinates is also multilinear, so ${\displaystyle \nabla ^{2}f=0}$ still holds when one or more variables are fixed. In other words, ${\displaystyle f}$ is harmonic on every "slice" of the domain along coordinate axes.

When the domain is rectangular in the coordinate axes (e.g. a hypercube), ${\displaystyle f}$ will have maxima and minima only on the vertices of the domain, i.e. the finite set of ${\displaystyle 2^{n}}$ points with minimal and maximal coordinate values. The value of the function on these points completely determines the function, since the value on the edges of the boundary can be found by linear interpolation, and the value on the rest of the boundary and the interior is fixed by Laplace's equation, ${\displaystyle \nabla ^{2}f=0}$.