The disconnectedness of certain sets defined after uni-variate polynomials

Year 2022, Volume: 5 Issue: 3, 119 - 133, 15.09.2022

Vladimir Kostov

https://doi.org/10.33205/cma.1111247

Cited By: 2

Abstract

We consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $d\geq 6$ and for signs of the coefficients $(+,-,+,+,\ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.

Keywords

Real polynomial in one variable, hyperbolic polynomial, Descartes’ rule of signs, discriminant set

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