Degree 6 Hyperbolic Polynomials and Orders of Moduli
Abstract
We consider real univariate degree d real-rooted polynomials with non-vanishing
coefficients. Descartes’ rule of signs implies that such a polynomial has ˜c positive and ˜p
negative roots counted with multiplicity, where ˜c and ˜p are the numbers of sign changes
and sign preservations in the sequence of its coefficients, ˜c + ˜p = d. For d = 6, we give
the exhaustive answer to the question: When the moduli of all 6 roots are distinct and
arranged on the real positive half-axis, in which positions can the moduli of the negative
roots be depending on the signs of the coefficients?
coefficients. Descartes’ rule of signs implies that such a polynomial has ˜c positive and ˜p
negative roots counted with multiplicity, where ˜c and ˜p are the numbers of sign changes
and sign preservations in the sequence of its coefficients, ˜c + ˜p = d. For d = 6, we give
the exhaustive answer to the question: When the moduli of all 6 roots are distinct and
arranged on the real positive half-axis, in which positions can the moduli of the negative
roots be depending on the signs of the coefficients?
Keywords
real polynomial in one variable; hyperbolic polynomial; sign pattern; Descartes’ rule of signs
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PDFISSN: 1331-0623 (Print), 1848-8013 (Online)