In algebra, a multilinear polynomial[1] is a multivariatepolynomial 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 2 or higher; that is, each monomial is a constant times a product of distinct variables. For example f(x,y,z) = 3xy + 2.5 y - 7z is a multilinear polynomial of degree 2 (because of the monomial 3xy) whereas f(x,y,z) = x² +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:
A multilinear polynomial is linear (affine) when varying only one variable, :
where and do not depend on . Note that is generally not zero, so is linear in the "shaped like a line" sense, but not in the "directly proportional" sense of a multilinear map.
More generally, every restriction of to a subset of its coordinates is also multilinear, so still holds when one or more variables are fixed. In other words, is harmonic on every "slice" of the domain along coordinate axes.
When the domain is rectangular in the coordinate axes (e.g. a hypercube), will have maxima and minima only on the vertices of the domain, i.e. the finite set of 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, .[1]
The value of the polynomial at an arbitrary point can be found by repeated linear interpolation along each coordinate axis. Equivalently, it is a weighted mean of the vertex values, where the weights are the Lagrange interpolation polynomials. These weights also constitute a set of generalized barycentric coordinates for the hyperrectangle. Geometrically, the point divides the domain into smaller hyperrectangles, and the weight of each vertex is the (fractional) volume of the hyperrectangle opposite it.
Algebraically, the multilinear interpolant on the hyperrectangle is:
where the sum is taken over the vertices . Equivalently,
where V is the volume of the hyperrectangle.
The value at the center is the arithmetic mean of the value at the vertices, which is also the mean over the domain boundary, and the mean over the interior. The components of the gradient at the center are proportional to the balance of the vertex values along each coordinate axis.
Multilinear polynomials are the interpolantsofmultilinearorn-linear interpolation on a rectangular grid, a generalization of linear interpolation, bilinear interpolation and trilinear interpolation to an arbitrary number of variables. This is a specific form of multivariate interpolation, not to be confused with piecewise linear interpolation. The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values.
where 
and 
do not depend on 
In other words, its Hessian matrix is a symmetric hollow matrix.
![{\displaystyle [a_{i},b_{i}]_{i=1}^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09dc7557550626b2e85c0794f48f482f7a1b657)
where the sum is taken over the vertices 
. Equivalently,
where V is the volume of the hyperrectangle.
