The disconnectedness of certain sets defined after uni-variate polynomials

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

References

1. A. Albouy, Y. Fu: Some remarks about Descartes’ rule of signs, Elem. Math., 69 (2014), 186-194.
2. B. Anderson, J. Jackson and M. Sitharam: Descartes’ rule of signs revisited, Am. Math. Mon., 105 (1998), 447-451.
3. V. I. Arnold: Hyperbolic polynomials and Vandermonde mappings, Funct. Anal. Appl., 20 (1986), 52-53.
4. 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, (1910) 171-215.
5. H. Cheriha, Y. Gati and V. P. Kostov: A nonrealization theorem in the context of Descartes’ rule of signs, Annual of Sofia University “St. Kliment Ohridski”, Faculty of Mathematics and Informatics, 106 (2019), 25-51.
6. H. Cheriha, Y. Gati and V. P. Kostov: Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs, Studia Scientiarum Mathematicarum Hungarica, 57 (2) (2020), 165-186.
7. H. Cheriha, Y. Gati and V. P. Kostov: On Descartes’ rule for polynomials with two variations of sign, Lithuanian Math. J., 60 (2020), 456-469.
8. H. Cheriha, Y. Gati and V. P. Kostov: Degree 5 polynomials and Descartes’ rule of signs, Acta Universitatis Matthiae Belii, series Mathematics, 28 (2020), 32-51.

9. D. R. Curtiss: Recent extensions of Descartes’ rule of signs, Ann. of Math., 19 (4) (1918), 251-278.
10. J. -P. de Gua de Malves: Démonstrations de la Règle de Descartes, Pour connoître le nombre des Racines positives & négatives dans les Équations qui n’ont point de Racines imaginaires, Memoires de Mathématique et de Physique tirés des registres de l’Académie Royale des Sciences (1741), 72-96.
11. R. Descartes: The Geometry of René Descartes: with a facsimile of the first edition, translated by D. E. Smith and M. L. Latham, Dover Publications, New York (1954).
12. J. Forsgård, V. P. Kostov and B. Shapiro: Could René Descartes have known this?, Exp. Math., 24 (4) (2015), 438-448.
13. J. Forsgård, V. P. Kostov and B. Shapiro: Corrigendum: "Could René Descartes have known this?”, Exp. Math., 28 (2) (2019), 255-256.
14. J. Forsgård, D. Novikov and B. Shapiro: A tropical analog of Descartes’ rule of signs, Int. Math. Res. Not., 12 (2017), 3726-3750.
15. 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 2, 291-309, Gauthier-Villars (1890).
16. C. F. Gauss: Beweis eines algebraischen Lehrsatzes, J. Reine Angew. Math., 3 (1-4) (1828); Werke 3, 67-70, Göttingen (1866).
17. A. B. Givental: Moments of random variables and the equivariant Morse lemma (Russian), Uspekhi Mat. Nauk, 42 (1987), 221-222.
18. D. J. Grabiner: Descartes’ Rule of Signs: Another Construction, Am. Math. Mon., 106 (1999), 854-856.
19. J. L. W. Jensen: Recherches sur la théorie des équations, Acta Math., 36 (1913), 181-195.
20. V. Jullien: Descartes La "Geometrie" de 1637.
21. V. P. Kostov: On the geometric properties of Vandermonde’s mapping and on the problem of moments. Proceedings of the Royal Society of Edinburgh, 112A, (1989), 203-211.
22. V. P. Kostov: On realizability of sign patterns by real polynomials, Czechoslovak Math. J., 68 (3) (2018), 143, 853–874.
23. V. P. Kostov: Polynomials, sign patterns and Descartes’ rule of signs, Math. Bohem., 144 (1) (2019), 39-67.
24. V. P. Kostov: Topics on hyperbolic polynomials in one variable. Panoramas et Synthèses 33, vi + 141 p. SMF (2011).
25. V. P. Kostov: Hyperbolic polynomials and canonical sign patterns, Serdica Math. J., 46 (2) (2020), 135-150.
26. V. P. Kostov: Univariate polynomials and the contractibility of certain sets, Annual of Sofia University “St. Kliment Ohridski”, Faculty of Mathematics and Informatics, 107 (2020), 75-99.
27. V. P. Kostov, B. Z. Shapiro: Polynomials, sign patterns and Descartes’ rule, Acta Universitatis Matthiae Belii, series Mathematics, 27 (2019), 1-11.
28. E. Laguerre: Sur la théorie des équations numériques, Journal de Mathématiques pures et appliquées, s. 3, 9, 1883, 99-146; oeuvres 1, Paris, 1898, Chelsea, New-York, 3–47 (1972).
29. I. Méguerditchian: Thesis - Géométrie du discriminant réel et des polynômes hyperboliques, thesis defended in 1991 at the University Rennes 1.
30. B. E. Meserve: Fundamental Concepts of Algebra, Dover Publications, New York (1982).