Multilinear polynomial

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. For example p(x,y,z) = 3xy + 2.5 y - 7z is a multilinear polynomial with degree 2 (because of the monomial 3xy) whereas p(x,y,z) = x^2 +4y is not.

They are important in the study of polynomial identity testing, because if a multilinear polynomial is zero on a set of vectors that span the space, it will be zero everywhere. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.^[1]

References

1. A. Giambruno, Mikhail Zaicev. Polynomial Identities and Asymptotic Methods. AMS Bookstore, 2005 ISBN 978-0-8218-3829-7. Section 1.3.