Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-20T09:16:12.813Z Has data issue: false hasContentIssue false
  • English
  • Français

On Lucas-Sets for Vector-Valued Abstract Polynomials in K-inner Product Spaces

Published online by Cambridge University Press:  20 November 2018

Neyamat Zaheer
Neyamat Zaheer*
Affiliation:
King Sand University, Riyadh, Saudi Arabia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The fact that Rolle's theorem on critical points of a real differentiable function does not hold in general for analytic functions of a complex variable raises a natural question [7, p. 21] as to whether or not it can be generalized to polynomials, the simplest subclass of analytic functions. While attempting to answer this and other related problems in [4, 5, 6], Lucas proved that all the critical points of a nonconstant polynomial f lie in the convex hull of the set of zeros of f (see Theorem (6.1) in [7]). Walsh [12] has shown that Lucas’ theorem is equivalent to the following result [7, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a polynomial f also contains all the zeros of its derivative f′.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 832 - 852
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

1. Bourbaki, N., Éléments de mathématique XIV. Livre II : Algèbre, Chap. VI : Groupes et corps ordonnés, Actualités Sci. Indust. 1179 (Hermann, Paris, 1952).Google Scholar
2. Hille, E. and Phillips, R. S., Functional analysis and semi-groups Amer. Math. Soc. Colloq. Publ. 31 (Amer. Math. Soc, Providence, R.I., 1957).Google Scholar
3. L., Hôrmander, On a theorem of Grace, Math. Scand. 2 (1954), 55–64.Google Scholar
4. Lucas, F., Propriétés géométriques des fractions rationelles, C.R. Acad. Sci. Paris 77 (1874), 431-433; ibid. 78 (1874), 140-144; ibid. 78 (1874), 180-183; ibid. 78 (1874), 271–274.Google Scholar
5. Lucas, F., Géométrie des polynômes, J. École Polytech. (1) 46 (1879), 1–33.Google Scholar
6. Lucas, F., Statique des polynômes, Bull. Soc. Math. France 17 (1888), 2–69.Google Scholar
7. Marden, M., Geometry of polynomials Math. Surveys 3 (Amer. Math. Soc, Providence, R.I., 1966).Google Scholar
8. Marden, M., A generalization of a theorem of Bôcher, SIAM J. Numer. Anal. 3 (1966), 269–275.Google Scholar
9. Marden, M., On composite abstract homogeneous polynomials, Proc. Amer. Math. Soc. 22 (1969), 28–33.Google Scholar
10. Taylor, A. E., Addition to the theory of polynomials in normed linear spaces, Tôhoku Math. J. 44 (1938), 302–318.Google Scholar
11. B. L., van der Waerden, Algebra, Vol. I, 4th éd., Die Grundlehren der Math. WissenschaftenSS (Springer-Verlag, Berlin, 1955).Google Scholar
12. Walsh, J. L., On the location of the roots of the derivative of a polynomial, in C.R. Congr. Internat, des Mathématiciens, Strasbourg, (1920), 339–342.Google Scholar
13. Wilansky, A., Functional analysis (Blaisdell, New York, 1964).Google Scholar
14. Zaheer, N., Null-sets of abstract homogeneous polynomials in vector spaces, Doctoral thesis, Univ. of Wisconsin, Milwaukee, Wisconsin (1971).Google Scholar
15. Zaheer, N., On polar relations of abstract homogeneous polynomials, Trans. Amer. Math. Soc. 281 (1976), 115–131.Google Scholar
16. Zaheer, N., On composite abstract homogeneous polynomials, Trans. Amer. Math. Soc. 228 (1977), 345–358.Google Scholar
17. Zaheer, N., On generalized polars of the product of abstract homogeneous polynomials, Pacific J. Math. 74 (1978), 535–557.Google Scholar
18. Zervos, S. P., Aspects modernes de la localisation des zéros des polynômes d'une variable, Ann. Sci. École Norm. Sup. 77 (1960), 303–410.Google Scholar