In algebra, a multilinear polynomial is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 that is linear

Linear

In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

 in each of its variables. In other words, no variable occurs to a power of 2 or higher; or alternatively, each monomial

Monomial

In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

 is a constant times a product of distinct variables. They are important in the study of polynomial identity testing, because if a multilinear polynomial is zero on a set of vectors that span

Linear span

In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

 the space, it will be zero everywhere. The degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.