A polynomial P(x1,⋯,xn) is
positive if its coefficients are ≧ 0 and not all 0, copositive if it is > 0 whenever the
variables are ≧ 0 and not all 0. Evidently (if needed, after multiplication by −1)
every real divisor Q of a positive polynomial P with P(0)≠0 is copositive.
Conversely, every real copositive polynomial Q with copositive highest and
copositive lowest homogeneous part is a divisor of a positive polynomial P, and
P∕Q may be chosen (1) as a product of positive 1-variable polynomials and
positive homogeneous polynomials; or, alternatively, (2) positive and so that
all terms of P from its lowest up to its highest degree are positive; or, if
n = 1, (3) so that P has no more nonzero terms than Q; or, for n = 1 and
quadratic Q, (4) as a power of a positive linear function, or (5) so that (1) and
(3) hold. For power series in a multidisk analogous results hold, for n = 1
partly depending on the finiteness of the number of complex zeros in the
disk.
