Abstract
The main topic of this chapter is the study of the zeros of polynomials with complex coefficients. This study leads to inequalities about the size of factors of polynomials. These inequalities play an important rôle in the last two chapters.
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References
P. Samuel, Théorie des Nombres Algébriques, Hermann, Paris, 1968, Appendix, Chapter 2.
Hermann, Paris, 1968, (Prob. 1, Ch. 2).
Sur quelques théorèmes de M. Petrovic relatifs aux zéros des fonctions analytiques; Bull. Soc. Math. France, 33, 1905, p. 251–261.
An inequality about factors of polynomials; Math. of Computation, 28, 1974, p. 1153–1157.
This inequality has been demonstrated in the author’s article — Entiers algébriques dont les conjugués sont proches du cercle unité, Seminaire Delange-Pisot-Poitou, 1977/78, exposé 39 — but with a more complicated method.
See, for example, the book of H. Dym and H.P. McKean, Fourier Series and Integrals, Academic Press, London, 1972.
M. Marden, The geometry of the zeros of a polynomial in a complex variable, American Mathematical Society, New York, 1949.
Computational complex analysis, Addison-Wesley, New York, 1976.
This method to compute the measure of a polynomial has been discovered independently and almost simultaneously by several people (including D. Boyd, M. Langevin, C. Stewart, and M. Mignotte).
Computing the measure of a polynomial, J. Symb. Comp., 4 (1) 1987, p. 21–34.
See, for example, G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934.
A.O. Gel’fond.— Transcendental and Algebraic Numbers, Dover, New York, 1960, Ch. 3, Sec. 4.
Preuss. Akad. Wiss. Sitzungsber., 1932, p. 321.
On the roots of certain algebraic equations, Proc. London Math. Soc., 33, 1931, p. 102–114.
Preuss. Akad. Wiss. Sitzungsber., 1933, p. 403–428.
On the distribution of the roots of polynomials, Ann. Math., 51, 1950, p. 105–119.
T. Ganelius, Sequences of analytic functions and their zeros, Arkiv för Math., 3, 1958, p. 1–50.
This construction has been published in the author’s article “Some Inequalities about univariate Polynomials”, Proceed. 1981 A.C.M. Symp. on Symbolic and Algebraic Computation, Snowbird, Utah.
K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation, Marcel Dekker, New York, 1974, Ch. 4, See. 2.
A.L. Cauchy, Exercices de Mathématiques, Quatrième Année, De Bure Frères, Paris, 1829. Oeuvres, Ser. II, Vol. IX, Gauthier-Villars, Paris, 1891.
M. Mignotte, An inequality on the greatest roots of a polynomial, Elemente der Mathematik, 1991.
Reference for this exercise: J.H. Davenport and M. Mignotte, On finding the largest root of a polynomial, MAN, 24, 6, 1990, p. 693–696.
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© 1992 Springer-Verlag New York, Inc.
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Mignotte, M. (1992). Polynomials with Complex Coefficients. In: Mathematics for Computer Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9171-5_4
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DOI: https://doi.org/10.1007/978-1-4613-9171-5_4
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