In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms of degree m such that Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, 2m = 2, or n = 3 and 2m = 4 a form is SOS if and only if it is positive. The same is also valid for the analog problem on positive symmetric forms. Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found. Moreover, every real nonnegative form can be approximated as closely as desired (in the l_1-norm of its coefficient vector) by a sequence of forms that are SOS.

Square matricial representation (SMR)

To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as where x^{{m}} is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying and L(\alpha) is a linear parameterization of the linear space The dimension of the vector x^{{m}} is given by whereas the dimension of the vector \alpha is given by Then, h(x) is SOS if and only if there exists a vector \alpha such that meaning that the matrix is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression was introduced in with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI. This representation is also known as Gram matrix.

Examples

h(x) is SOS.

Generalizations

Matrix SOS

A matrix form F(x) (i.e., a matrix whose entries are forms) of dimension r and degree 2m in the real n-dimensional vector x is SOS if and only if there exist matrix forms of degree m such that

Matrix SMR

To establish whether a matrix form F(x) is SOS amounts to solving a convex optimization problem. Indeed, similarly to the scalar case any F(x) can be written according to the SMR as where \otimes is the Kronecker product of matrices, H is any symmetric matrix satisfying and L(\alpha) is a linear parameterization of the linear space The dimension of the vector \alpha is given by Then, F(x) is SOS if and only if there exists a vector \alpha such that the following LMI holds: The expression was introduced in in order to establish whether a matrix form is SOS via an LMI.

Noncommutative polynomial SOS

Consider the free algebra R⟨X⟩ generated by the n noncommuting letters X = (X1, ..., Xn) and equipped with the involution T, such that T fixes R and X1, ..., Xn and reverses words formed by X1, ..., Xn. By analogy with the commutative case, the noncommutative symmetric polynomials f are the noncommutative polynomials of the form f = fT . When any real matrix of any dimension r × r is evaluated at a symmetric noncommutative polynomial f results in a positive semi-definite matrix, f is said to be matrix-positive. A noncommutative polynomial is SOS if there exists noncommutative polynomials such that Surprisingly, in the noncommutative scenario a noncommutative polynomial is SOS if and only if it is matrix-positive. Moreover, there exist algorithms available to decompose matrix-positive polynomials in sum of squares of noncommutative polynomials.