Polynomials with Nonnegative Coefficients

Abstract

Abstract.The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independently by B. Conroy in connection with his work in number theory. A crucial and interesting lemma is proved which describes general coefficient-root relations for polynomials with nonnegative coefficients and for polynomials for which the case of equality holds in Descarte's Rule of Signs.

Progress on the unfair 0-1-polynomials conjecture using linear recurrences and numerical analysis

“…Finally, it is in order to mention some works on factorizations with constrained coefficients. Indeed, intuitively, it would make sense that some progress on the conjecture would follow from a study of the divisors and the roots of 0-1-polynomials (Newman polynomials) [19][20][21][22][23][24][25][26] and of polynomials with nonnegative coefficients [27][28][29][30]. Variations on this theme include the study of polynomials with coefficients that belong to a finite set [31] or that are constrained by inequalities [32].…”