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polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem
Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.
[1] C. F. Gauss: Collected Works. Leipzig, Teubner, 1900–1903, vol. 3, p. 112, and vol. 8, p. 32, and vol. 9, p. 187. Zbl 0924.01032
[2] F. Lucas: Propriétés géométriques des fractions rationelles. C. R. Acad. Sci. Paris 77 (1874), 431–433.
[3] J. L. Díaz-Barrero: Characterization of Polynomials by Reflection Coefficients. PhD. Disertation (Advisor J. J. Egozcue), Universitat Politècnica de Catalunya, Barcelona, 2000.
[4] M. Marden: The Geometry of the Zeros of a Polynomial in a Complex Variable. American Mathematical Society, Rhode Island, 1966. MR 0031114
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