A contraction of the Lucas polygon
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- by Branko Ćurgus and Vania Mascioni
- Proc. Amer. Math. Soc. 132 (2004), 2973-2981
- DOI: https://doi.org/10.1090/S0002-9939-04-07231-4
- Published electronically: May 20, 2004
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Abstract:
The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial $p$ lie in the convex hull of the roots of $p$, called the Lucas polygon of $p$. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of $p’$ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon of $p$.References
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- Branko Ćurgus and Vania Mascioni, On the location of critical points of polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 1, 253–264. MR 1929045, DOI 10.1090/S0002-9939-02-06534-6
- Dimitar K. Dimitrov, A refinement of the Gauss-Lucas theorem, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2065–2070. MR 1452801, DOI 10.1090/S0002-9939-98-04381-0
- Peter Henrici, Applied and computational complex analysis. Vol. 1, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Power series—integration—conformal mapping—location of zeros; Reprint of the 1974 original; A Wiley-Interscience Publication. MR 1008928
- Einar Hille, Analytic function theory. Vol. 1, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass., 1959. MR 0107692
- M. Marden: The Location of the Zeros of the Derivative of a Polynomial, American Mathematical Monthly, 42 (1935), 277–286.
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992. Translated from the French by Catherine Mignotte. MR 1140923, DOI 10.1007/978-1-4613-9171-5
- Maurice Mignotte and Doru Ştefănescu, Polynomials, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer-Verlag Singapore, Singapore; Centre for Discrete Mathematics & Theoretical Computer Science, Auckland, 1999. An algorithmic approach. MR 1690362
- G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187, DOI 10.1142/1284
- Andrzej Turowicz, Geometria zer wielomianów, Państwowe Wydawnictwo Naukowe, Warsaw, 1967 (Polish). MR 0229806
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208
Bibliographic Information
- Branko Ćurgus
- Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
- Email: curgus@cc.wwu.edu
- Vania Mascioni
- Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306-0490
- Email: vdm@bsu-cs.bsu.edu
- Received by editor(s): October 29, 2002
- Received by editor(s) in revised form: February 12, 2003
- Published electronically: May 20, 2004
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2973-2981
- MSC (2000): Primary 30C15; Secondary 26C10
- DOI: https://doi.org/10.1090/S0002-9939-04-07231-4
- MathSciNet review: 2063118