Summary
The present paper deals with certain new aspects of the theory of algebra-valued generalized polars of the product of abstract homogeneous polynomials, opening up new avenues to the existing (and still very young) theory put forward by Marden and Zaheer (see Pacific J. Math., 74 (1978), n. 2, pp. 535–557). Our main theorem here leads, on the one hand, to an improved version of Walsh's cross-ratio theorem on critical points of functions of the formf 1 f 2/f 3 (f i being complex-valued polynomials) and, on the other hand, it offers a two-fold generalization of one of the main theorems of Zaheer in the paper cited above.
Article PDF
Similar content being viewed by others
References
M. Bôcher,A problem in statics and its relation to certain algebraic invariants, Proc. Amer. Acad. Arts Sci.,40 (1904), pp. 469–484.
N. Bourbaki,Éléments de mathématique. XIV. Part 1,Les structures fondamentales de l'analyse. Livre II:Algèbre. Chap. VI:Groupes et corps ordonnés (Actualités Sci. Indust., n. 1179, Hermann, Paris, 1952).
J. H. Grace,On the zeros of a polynomial, Proc. Cambridge Philos. Soc.,14 (1900–1902), pp. 352–357.
E. Hille -R. S. Phillips,Functional analysis and semigroups, rev. ed. (Amer. Math. Soc. Colloq. Publ., Vol. 31, Providence, R.I., 1957).
L. Hörmander,On a theorem of Grace, Math. Scand.,2 (1954), pp. 55–64.
E. Laguerre,Oeuvres de Laguerre, Vol. 1:Algèbre. Calcul Intégral, Gauthier-Villars, Paris, 1898.
M. Marden,Geometry of polynomials, rev. ed. (Math. Surveys, n. 3, Amer. Math. Soc., Providence, R. I., 1966).
M. Marden,A generalization of a theorem of Bôeher, SIAM J. Numer. Anal.,3 (1966), pp. 269–275.
M. Marden,On composite abstract homogeneous polynomials, Proc. Amer. Math. Soc.,22 (1969), pp. 28–33.
A. E. Taylor,Addition to the theory of polynomials, Tôhoku. Math. J.,44 (1938), pp. 302–318.
S. Szegö,Bemerkungen au einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z.,13 (1922), pp. 28–55.
B. L. Van der Waerden,Algebra, Vol. I, 4-th ed.,Die Grundlehren der math. Wissenschaften, Band33 (Berlin, Springer-Verlag, 1955), English transl. (New York, Ungar, 1970).
J. L. Walsh,On the location of the roots of the Jacobian of two binary forms, and of the derivative of a rational function, Trans. Amer, Math. Soc.,22 (1921), pp. 101–116.
J. L. Waish,Sur la position des racines des dérivées d'un polynôme, C. R. Acad. Sci. Paris,172 (1921), pp. 662–664.
A. Wilansky,Functional analysis (New York, Blaisdell, 1964).
N. Zaheer,Null-sets of abstract homogeneous polynomials in vector spaces, Ph. D. Thesis (University of Wisconsin, Milwaukee, Wisc., 1971).
N. Zaheer,On polar relations of abstract homogeneous polynomials, Trans. Amer. Math. Soc.,218 (1976), pp. 115–131.
N. Zaheer,On composite abstract homogeneous polynomials, Trans. Amer. Math. Soc.,228 (1977), pp. 345–358.
N. Zaheer,On generalized polars of the product of abstract homogeneous polynomials, Pacific J. Math.,74 (1978), n. 2, pp. 535–557.
N.Zaheer,On the theory of algebra-valued generalized polars, Indiana Univ. Math. J. (To appear in September-October 1980).
N. Zaheer -M. Alam,Zeros of polar-composite polynomials in algebraically closed fields, Proc. Ind. Math. Soc., (3),40 (1980), pp. 527–552.
S. P. Zervos,Aspects modernes de la localisation des zeros des polynômes d'une variable, Ann. Sci. École Norm. Sup., (3),77 (1960), pp. 303–410.
Author information
Authors and Affiliations
Additional information
Supported by the UGC Fellowship, Government of India.
Rights and permissions
About this article
Cite this article
Zaheer, N., Khan, A.A. Cross-ratio theorems on generalized polars of abstract homogeneous polynomials. Annali di Matematica pura ed applicata 126, 363–377 (1980). https://doi.org/10.1007/BF01762516
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762516