Geodesic
On realizability of sign patterns by real polynomials
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 853-874 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/CMJ.2018.0163-17
Classification : 26C10, 30C15
Keywords: real polynomial in one variable; sign pattern; Descartes' rule of signs 
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     title = {On realizability of sign patterns by real polynomials},
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Kostov, Vladimir. On realizability of sign patterns by real polynomials. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 853-874. doi: 10.21136/CMJ.2018.0163-17

[1] Albouy, A., Fu, Y.: Some remarks about Descartes' rule of signs. Elem. Math. 69 (2014), 186-194. | DOI | MR | JFM

[2] Anderson, B., Jackson, J., Sitharam, M.: Descartes' rule of signs revisited. Am. Math. Mon. 105 (1998), 447-451. | DOI | MR | JFM

[3] Forsgård, J., Kostov, V. P., Shapiro, B.: Could René Descartes have known this?. Exp. Math. 24 (2015), 438-448. | DOI | MR | JFM

[4] Grabiner, D. J.: Descartes' rule of signs: another construction. Am. Math. Mon. 106 (1999), 845-856. | DOI | MR | JFM

[5] Kostov, V. P.: Topics on Hyperbolic Polynomials in One Variable. Panoramas et Synthèses 33, Société Mathématique de France (SMF), Paris (2011). | MR | JFM

[6] Shapiro, B. Z., Khesin, B. A.: Swallowtails and Whitney umbrellas are homeomorphic. J. Algebr. Geom. 1 (1992), 549-560. | MR | JFM

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