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Keywords:
real polynomial in one variable; sign pattern; Descartes' rule of signs
real polynomial in one variable; sign pattern; Descartes' rule of signs
Summary:
The classical Descartes' rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p,n)$, chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials.
References:
[1] Albouy, A., Fu, Y.: Some remarks about Descartes' rule of signs. Elem. Math. 69 (2014), 186-194. DOI 10.4171/EM/262 | MR 3272179 | Zbl 1342.12002
[2] Anderson, B., Jackson, J., Sitharam, M.: Descartes' rule of signs revisited. Am. Math. Mon. 105 (1998), 447-451. DOI 0913.12001 | MR 1622513 | Zbl 0913.12001
[3] Forsgård, J., Kostov, V. P., Shapiro, B.: Could René Descartes have known this?. Exp. Math. 24 (2015), 438-448. DOI 10.1080/10586458.2015.1030051 | MR 3383475 | Zbl 1326.26027
[4] Grabiner, D. J.: Descartes' rule of signs: another construction. Am. Math. Mon. 106 (1999), 845-856. DOI 10.2307/2589619 | MR 1732666 | Zbl 0980.12001
[5] Kostov, V. P.: Topics on Hyperbolic Polynomials in One Variable. Panoramas et Synthèses 33, Société Mathématique de France (SMF), Paris (2011). MR 2952044 | Zbl 1259.12001
[6] Shapiro, B. Z., Khesin, B. A.: Swallowtails and Whitney umbrellas are homeomorphic. J. Algebr. Geom. 1 (1992), 549-560. MR 1174901 | Zbl 0790.57019
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