Problems on algebraical polynomials appear in many fields of mathematics and computer science. Especially the task of determining the roots of polynomials has been frequently investigated.Nonetheless, the task of locating the zeros of complex polynomials is still challenging. In this paper we deal with the location of zeros of univariate complex polynomials. We prove some novel upper bounds for the moduli of the zeros of complex polynomials. That means, we provide disks in the complex plane where all zeros of a complex polynomial are situated. Such bounds are extremely useful for obtaining a priori assertations regarding the location of zeros of polynomials. Based on the proven bounds and a test set of polynomials, we present an experimental study to examine which bound is optimal.
<p>[1] R. E Allerdice, On a limit of the roots of an equation that is independent of all but two of the coefficients. Bull. Amer. Math. Soc. Vol. 13, 1906/07, 443-447.
[2] H. Anai, K. Horimoto, Algebraic Biology 2005. Proceedings of the First International Conference on Algebraic Biology. Computer Algebra in Biology, 2005, Tokyo/Japan.
[3] C. Bissel, Control Engineering. 2nd Edition, CRC Press, 1996.
[4] P. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities. Springer, 1995.
[5] J. E. Brown, G. Xiang, Proof of the Sendov conjecture for polynomials of degree at most eight. Journal of Mathematical Analysis and Applications, Vol. 232, No2, 1999, 272-292.
[6] A. Byrne, Some results for the Sendov conjecture. Journal of Mathematical Analysis and Applications, Vol. 199, No3, 1996, 754-768.
[7] M. Dehmer, On the location of zeros of complex polynomials. Journal of Inequalities in Pure and Applied Mathematics, Vol. 7, Issue 1, 2006.
[8] M. Dehmer, Die analytische Theorie der Polynome. Nullstellenschranken fur komplexwertige Polynome. Weissensee-Verlag, Berlin, 2004.
[9] J. Dieudonne, La theorie des polynomes d-une variable. Memor. Sci. Math., Vol. 93, 1938.
[10] B. Fine, G. Rosenberger, The Fundamental Theorem of Algebra. Springer, 1997. [11] W. Heitzinger W. I. Troch, G. Valentin, Praxis nichtlinearer Gleichungen. Carl Hanser Verlag, M¨unchen-Wien, 1985.
[12] P. Henrici, Applied and computational complex analysis. Volume I. Wiley Classics Library Edition, 1988.
[13] M. Marden, Geometry of polynomials. Mathematical Surveys. Amer. Math. Soc., Rhode Island, Vol. 3, 196.
[14] M. Mignotte, D. Stefanescu, Estimates for polynomial roots. Appl. Alg. Eng. Comm. Comp., Vol. 12, 2001, 437-453.
[15] M. Mignotte, D. Stefanescu, Polynomials: An Algorithmic Approach. Springer Series in Discrete Mathematics and Theoretical Computer Science, 1999
[16] G. V. Milovanovic, D. S. Mitrinovic, T. M. Rassias, Topics in Polynomials. Extremal Problems, Inequalities, Zeros. World Scientific, 1994
[17] N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome. Hochschulb¨ucher für Mathematik. Vol. 55. Berlin, VEB Deutscher Verlag der Wissenschaften, 196. [18] R. Pereira, Differentiators and the geometry of polynomials. J. Math. Anal. Appl. 285, Vol. 1, 2003, 336-348.
[19] R. Pereira, Trace Vectors in Matrix Analysis. PhD thesis, University of Toronto, 2003.
[20] V. V. Prasolov, Polynomials. Springer, 2004.
[21] Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials. Critical Points, Zeros and Extremal Properties. Clarendon Press, 2002.
[22] F. R├╝hs, Funktionentheorie. VEB Deutscher Verlag der Wissenschaften Berlin, 1971.
[23] H. Sagan, Boundary and Eigenvalue Problems in Mathematical Physics. Dover Publications, 1989.
[24] W. Specht, Die Lage der Nullstellen eines Polynoms. Mathematische Nachrichten, Vol. 15, 1956, 353-374.
[25] U. Zölzer, Digital Audio Signal Processing. John Wiley & Sons, 1997.</p>
<p>[1] R. E Allerdice, On a limit of the roots of an equation that is independent of all but two of the coefficients. Bull. Amer. Math. Soc. Vol. 13, 1906/07, 443-447.
[2] H. Anai, K. Horimoto, Algebraic Biology 2005. Proceedings of the First International Conference on Algebraic Biology. Computer Algebra in Biology, 2005, Tokyo/Japan.
[3] C. Bissel, Control Engineering. 2nd Edition, CRC Press, 1996.
[4] P. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities. Springer, 1995.
[5] J. E. Brown, G. Xiang, Proof of the Sendov conjecture for polynomials of degree at most eight. Journal of Mathematical Analysis and Applications, Vol. 232, No2, 1999, 272-292.
[6] A. Byrne, Some results for the Sendov conjecture. Journal of Mathematical Analysis and Applications, Vol. 199, No3, 1996, 754-768.
[7] M. Dehmer, On the location of zeros of complex polynomials. Journal of Inequalities in Pure and Applied Mathematics, Vol. 7, Issue 1, 2006.
[8] M. Dehmer, Die analytische Theorie der Polynome. Nullstellenschranken fur komplexwertige Polynome. Weissensee-Verlag, Berlin, 2004.
[9] J. Dieudonne, La theorie des polynomes d-une variable. Memor. Sci. Math., Vol. 93, 1938.
[10] B. Fine, G. Rosenberger, The Fundamental Theorem of Algebra. Springer, 1997. [11] W. Heitzinger W. I. Troch, G. Valentin, Praxis nichtlinearer Gleichungen. Carl Hanser Verlag, M¨unchen-Wien, 1985.
[12] P. Henrici, Applied and computational complex analysis. Volume I. Wiley Classics Library Edition, 1988.
[13] M. Marden, Geometry of polynomials. Mathematical Surveys. Amer. Math. Soc., Rhode Island, Vol. 3, 196.
[14] M. Mignotte, D. Stefanescu, Estimates for polynomial roots. Appl. Alg. Eng. Comm. Comp., Vol. 12, 2001, 437-453.
[15] M. Mignotte, D. Stefanescu, Polynomials: An Algorithmic Approach. Springer Series in Discrete Mathematics and Theoretical Computer Science, 1999
[16] G. V. Milovanovic, D. S. Mitrinovic, T. M. Rassias, Topics in Polynomials. Extremal Problems, Inequalities, Zeros. World Scientific, 1994
[17] N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome. Hochschulb¨ucher für Mathematik. Vol. 55. Berlin, VEB Deutscher Verlag der Wissenschaften, 196. [18] R. Pereira, Differentiators and the geometry of polynomials. J. Math. Anal. Appl. 285, Vol. 1, 2003, 336-348.
[19] R. Pereira, Trace Vectors in Matrix Analysis. PhD thesis, University of Toronto, 2003.
[20] V. V. Prasolov, Polynomials. Springer, 2004.
[21] Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials. Critical Points, Zeros and Extremal Properties. Clarendon Press, 2002.
[22] F. R├╝hs, Funktionentheorie. VEB Deutscher Verlag der Wissenschaften Berlin, 1971.
[23] H. Sagan, Boundary and Eigenvalue Problems in Mathematical Physics. Dover Publications, 1989.
[24] W. Specht, Die Lage der Nullstellen eines Polynoms. Mathematische Nachrichten, Vol. 15, 1956, 353-374.
[25] U. Zölzer, Digital Audio Signal Processing. John Wiley & Sons, 1997.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54714", author = "Matthias Dehmer1 Jürgen Kilian", title = "On Bounds For The Zeros of Univariate Polynomial", abstract = "Problems on algebraical polynomials appear in many fields of mathematics and computer science. Especially the task of determining the roots of polynomials has been frequently investigated.Nonetheless, the task of locating the zeros of complex polynomials is still challenging. In this paper we deal with the location of zeros of univariate complex polynomials. We prove some novel upper bounds for the moduli of the zeros of complex polynomials. That means, we provide disks in the complex plane where all zeros of a complex polynomial are situated. Such bounds are extremely useful for obtaining a priori assertations regarding the location of zeros of polynomials. Based on the proven bounds and a test set of polynomials, we present an experimental study to examine which bound is optimal.
", keywords = "complex polynomials, zeros, inequalities", volume = "1", number = "2", pages = "137-6", }
