Vol. 29, No. 3, 1969

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Divisors of polynomials and power series with positive coefficients

T. S. Motzkin and Ernst Gabor Straus

Vol. 29 (1969), No. 3, 641–652

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.

Mathematical Subject Classification
Primary: 32.10
Secondary: 26.00
Received: 15 October 1968
Published: 1 June 1969
T. S. Motzkin
Ernst Gabor Straus