Mathematica Bohemica, Vol. 144, No. 1, pp. 39-67, 2019
Polynomials, sign patterns and Descartes' rule of signs
Vladimir Petrov Kostov
Received August 16, 2017. Published online June 18, 2018.
Abstract: By Descartes' rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has ${\rm pos}\leq c$ positive and $\neg\leq p$ negative roots, where ${\rm pos}\equiv c\pmod2$ and $\neg\equiv p\pmod2$. For $1\leq d\leq3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $({\rm pos}, {\rm neg})$ satisfying these conditions there exists a polynomial $P$ with exactly ${\rm pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\geq4$ this is not so. It was observed that for $4\leq d\leq8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\geq4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair.
Keywords: real polynomial in one variable; sign pattern; Descartes' rule of signs
References: [1] A. Albouy, Y. Fu: Some remarks about Descartes' rule of signs. Elem. Math. 69 (2014), 186-194. DOI 10.4171/EM/262 | MR 3272179 | Zbl 1342.12002
[2] F. Cajori: A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity. Colorado College Publication, Science Series 12 (1910), 171-215, 217-287. JFM 42.0057.02
[3] J. Forsgard, V. P. Kostov, B. Z. Shapiro: Could René Descartes have known this? Exp. Math. 24 (2015), 438-448. DOI 10.1080/10586458.2015.1030051 | MR 3383475 | Zbl 1326.26027
[4] J. Forsgard, V. P. Kostov, B. Z. Shapiro: Corrigendum:"Could René Descartes have known this?". To appear in Exp. Math. DOI 10.1080/10586458.2017.1417775
[5] J. Fourier: Sur l'usage du théorème de Descartes dans la recherche des limites des racines. Bulletin des sciences par la Société Philomatique de Paris (1820), 156-165, 181-187; Oeuvres de Fourier publiées par les soins de M. Gaston Darboux sous les auspices du ministère de l'instruction publique. Tome II. Mémoires publiés dans divers recueils Gauthier-Villars, Paris (1890), 291-309 (In French.). JFM 22.0021.01
[6] C. F. Gauss: Beweis eines algebraischen Lehrsatzes. J. Reine Angew. Math. 3 (1828), 1-4 (In German.). DOI 10.1515/crll.1828.3.1 | MR 1577673 | Zbl 003.0089cj
[7] D. J. Grabiner: Descartes' rule of signs: Another construction. Am. Math. Mon. 106 (1999), 854-856. DOI 10.2307/2589619 | MR 1732666 | Zbl 0980.12001
[8] V. P. Kostov: On realizability of sign patterns by real polynomials. To appear in Czech. Math. J. DOI 10.21136/CMJ.2018.0163-17
[9] V. P. Kostov, B. Shapiro: Something you always wanted to know about real polynomials (but were afraid to ask). Avaible at https://arxiv.org/abs/1703.04436.
Affiliations: Vladimir Petrov Kostov, Université Côte d'Azur, Le Centre national de la recherche scientifique (CNRS), Laboratoire Jean-Alexandre Dieudonné (LJAD), 28 Avenue de Valrose, 06108 Nice CEDEX 2, France, e-mail: vladimir.kostov@unice.fr
