ABSENCE OF POSITIVE ROOTS OF SEXTIC POLYNOMIALS

Abstract

Given a general monic sextic polynomial with six real coefficients, necessary and sufficient conditions are found such that the polynomial does not have any positive roots. This `nonlinear eigenvalue problem' is a relatively difficult one since we have $6$ real parameters. Fortunately, we succeed in applying the Cheng-Lin envelope method in [1] together with several new ideas and techniques to express our criteria in terms of roots of quartic polynomials and explicit parametric curves and therefore our problem is completely solved. Several specific examples are also included to illustrate various applications including the seeking of periodic solutions of the logistic equation studied in chaos theory.