  We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that
are the sum of univariate polynomials in different variables and a quadratic
polynomial. Motivated by the fact that nonnegative separable and nonnegative
quadratic polynomials are sums of squares, we study whether nonnegative SPQ
polynomials are (i) the sum of a nonnegative separable and a nonnegative
quadratic polynomial, and (ii) a sum of squares. We establish that the answer
to question (i) is positive for univariate plus quadratic polynomials and for
convex SPQ polynomials, but negative already for bivariate quartic SPQ
polynomials. We use our decomposition result for convex SPQ polynomials to show
that convex SPQ polynomial optimization problems can be solved by "small"
semidefinite programs. For question (ii), we provide a complete
characterization of the answer based on the degree and the number of variables
of the SPQ polynomial. We also prove that testing nonnegativity of SPQ
polynomials is NP-hard when the degree is at least four. We end by presenting
applications of SPQ polynomials to upper bounding sparsity of solutions to
linear programs, polynomial regression problems in statistics, and a
generalization of Newton's method which incorporates separable higher-order
derivative information.
