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 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables. For example f ( x , y , z ) = 3 x y + 2.5 y − 7 z {\displaystyle f(x,y,z)=3xy+2.5y-7z} is a multilinear polynomial of degree 2 {\displaystyle 2} (because of the monomial 3 x y {\displaystyle 3xy} ) whereas f ( x , y , z ) = x 2 + 4 y {\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.
Definition 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: f ( x ) = ∑ i 1 = 0 1 ∑ i 2 = 0 1 ⋯ ∑ i n = 0 1 a i 1 i 2 ⋯ i n x 1 i 1 x 2 i 2 ⋯ x n i n {\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: f ( x , y ) = ∑ i = 0 1 ∑ j = 0 1 a i j x i y j = a 00 + a 10 x + a 01 y + a 11 x y = ( 1 x ) ( a 00 a 01 a 10 a 11 ) ( 1 y ) {\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}}}
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