  Lump solutions are analytical rational function solutions localized in all
directions in space. We analyze a class of lump solutions, generated from
quadratic functions, to nonlinear partial differential equations. The basis of
success is the Hirota bilinear formulation and the primary object is the class
of positive multivariate quadratic functions. A complete determination of
quadratic functions positive in space and time is given, and positive quadratic
functions are characterized as sums of squares of linear functions. Necessary
and sufficient conditions for positive quadratic functions to solve Hirota
bilinear equations are presented, and such polynomial solutions yield lump
solutions to nonlinear partial differential equations under the dependent
variable transformations u=2(ln f)_x and u=2(ln f)_{xx}, where x is one spatial
variable. Applications are made for a few generalized KP and BKP equations.
