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Polynomials with non-negative coefficients



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Polynomials with non negative coefficients R W Barnard W Dayawansa D Weinberg K Pearce Department of Mathematics Texas Tech University Lubbock TX 79409 1 Introduction Can a conjugate pair of zeros be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients This question has attracted considerable attention during the last few years because of its seemingly elementary nature and its potential for applications in number theory and control theory A proposed answer to this question arose as a conjecture out of the work in 6 where explicit bounds were determined for the constants occurring in the Beauzamy Enflo generalization 1 2 of Jensen s Inequality An affirmative answer to this question was conjectured independently by B Conrey in connection with some of his work in number theory Conrey announced the conjecture at the annual West Coast Number Theory Conference held in December 1987 The main theorem in this note gives a positive answer to the question This will be proved in section 2 The principle ingredients of the proof are an idea from index theory classical properties of polynomials and a significant lemma Lemma 2 1 which we prove in section 3 This lemma states some strong consequences for the case of equality holding in Descartes Rule of Signs see 5 We also prove a corollary of the main theorem which describes the region into which certain zeros can be moved while preserving the nonnegativity of the coefficients Formerly at Texas Tech University Currently in the Department of Electrical Engineering University of Maryland College Park Md 20742 1 Consider the polynomial defined by PN z 1 z N We note that PN z N 1 Y 1 l 0 z i N 2 l N e 2 1 2 cos 0 z z N 2 X bk z k 1 k 0 has bk 0 0 k N 2 if 0 N while any other choice for 0 produces a factor with some negative bk coefficients Thus one initial suggestion for the general question of factoring out a conjugate pair of zeros was to factor out a pair of



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