@article{ZNSL_2009_373_a0,
author = {A. G. Akritas},
title = {Vincent's theorem of~1836: overview and future research},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {5--33},
year = {2009},
volume = {373},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a0/}
}
A. G. Akritas. Vincent's theorem of 1836: overview and future research. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 5-33. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a0/
[1] A. G. Akritas, Vincent's theorem in algebraic manipulation, Ph. D. Thesis, Operations Research Program, North Carolina State University, Raleigh, NC, 1978
[2] A. G. Akritas, “An implementation of Vincent's Theorem”, Numerische Mathematik, 36 (1980), 53–62 | DOI | MR | Zbl
[3] A. G. Akritas, “The fastest exact algorithms for the isolation of the real roots of a polynomial equation”, Computing, 24 (1980), 299–313 | DOI | MR | Zbl
[4] A. G. Akritas, “Reflections on a pair of theorems by Budan and Fourier”, Mathematics Magazine, 55:5 (1982), 292–298 | DOI | MR | Zbl
[5] A. G. Akritas, “There is no ‘Uspensky’s method' ”, Proceedings of the 1986 Symposium on Symbolic and Algebraic Computation, Waterloo, Ontario, Canada, 1986, 88–90
[6] A. G. Akritas, Elements of Computer Algebra with Applications, John Wiley Interscience, New York, 1989 | MR | Zbl
[7] A. G. Akritas, “There is no ‘Descartes’ method' ”, Computer Algebra in Education, eds. M. J. Wester, M. Beaudin, AullonaPress, USA, 2008, 19–35
[8] A. G. Akritas, “Linear and quadratic complexity bounds on the values of the positive roots of polynomials” (to appear)
[9] A. G. Akritas , A. I, Argyris, A. W. Strzeboński, “FLQ, the Fastest Quadratic Complexity Bound on the Values of Positive Roots of Polynomials”, Serdica Journal of Computing, 2 (2008), 145–162 | MR | Zbl
[10] A. G. Akritas, A. Bocharov, A. W. Strzeboński, “Implementation of real root isolation algorithms in Mathematica”, Abstracts of the International Conference on Interval and Computer-Algebraic Methods in Science and Engineering, Interval' 94 (St. Petersburg, Russia, March 7–10, 1994), 23–27
[11] A. G. Akritas, S. D. Danielopoulos, “On the forgotten theorem of Mr. Vincent”, Historia Mathematica, 5 (1978), 427–435 | DOI | MR | Zbl
[12] A. G. Akritas, S. D. Danielopoulos, “An unknown theorem for the isolation of the roots of polynomials”, Ganita-Bharati. Bulletin of the Indian Society for History of Mathematics, 2, 1980, 41–49 | MR | Zbl
[13] A. G. Akritas, K. H. Ng, “Exact algorithms for polynomial real root approximation using continued fractions”, Computing, 30 (1983), 63–76 | DOI | MR | Zbl
[14] A. G. Akritas, A. Strzeboński, “A comparative study of two real root isolation methods”, Nonlinear Analysis: Modelling and Control, 10:4 (2005), 297–304 | MR | Zbl
[15] A. G. Akritas, P. Vigklas, “A Comparison of Various Methods for Computing Bounds for Positive Roots of Polynomials”, Journal of Universal Computer Science, 13:4 (2007), 455–467 | MR
[16] A. G. Akritas, A. Strzeboński, P. Vigklas, “Implementations of a New Theorem for Computing Bounds for Positive Roots of Polynomials”, Computing, 78 (2006), 355–367 | DOI | MR | Zbl
[17] A. G. Akritas, A. Strzeboński, P. Vigklas, “Advances on the Continued Fractions Method Using Better Estimations of Positive Root Bounds”, Proceedings of the 10th International Workshop on Computer Algebra in Scientific Computing, CASC–2007 (Bonn, Germany, September 16–20, 2007), LNCS, 4770, eds. V. G. Ganzha, E. W. Mayr, E. V. Vorozhtsov, Springer Verlag, Berlin, 24–30 | Zbl
[18] A. G. Akritas, A. Strzeboński, P. Vigklas, “On the Various Bisection Methods Derived from Vincent's Theorem”, Serdica Journal of Computing, 2 (2008), 89–104 | MR | Zbl
[19] A. G. Akritas, A. Strzeboński, P. Vigklas, “Improving the Performance of the Continued Fractions Method Using new Bounds of Positive Roots”, Nonlinear Analysis: Modelling and Control, 13:3 (2008), 265–279 | MR | Zbl
[20] A. Alesina, M. Galuzzi, “A new proof of Vincent's theorem”, L'Enseignement Mathemathique, 44 (1998), 219–256 | MR | Zbl
[21] A. Alesina, M. Galuzzi, “Addentum to the paper ‘A new proof of Vincent’s theorem' ”, L'Enseignement Mathemathique, 45 (1999), 379–380 | MR
[22] A. Alesina, M. Galuzzi, “Vincent's Theorem from a Modern Point of View”, Categorical Studies in Italy 2000, Rendiconti del Circolo Matematico di Palermo Serie II, 64, eds. Betti R., Lawvere W. F., 2000, 179–191 | MR | Zbl
[23] E. Bombieri, A. J. van der Poorten, “Continued fractions of algebraic numbers”, Computational Algebra and Number Theory (Sydney, 1992), Math. Appl., 325, Kluwer Academic Publishers, Dordrecht, 1995, 137–152 | MR | Zbl
[24] F. Boulier, Systèmes polynomiaux: que signifie “résoudre”?, Lect. Notes, Université Lille 1, 8 janvier 2007; ; http://www2.lifl.fr/~boulier/RESOUDRE/SHARED/support.pdfhttp://www.fil.univ-lille1.fr/portail/ls4/resoudre
[25] F. Boulier, Private Communication, October 2007
[26] D. G. Cantor, P. H. Galyean, H. G. Zimmer, “A Continued Fraction Algorithm for Real Algebraic Numbers”, Mathematics of Computation, 26:119 (1972), 785–791 | DOI | MR | Zbl
[27] G. E. Collins, A. G. Akritas, “Polynomial real root isolation using Descartes' rule of signs”, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, Yorktown Heights, N.Y., 1976, 272–275
[28] H. Hong, “Bounds for absolute positiveness of multivariate polynomials”, J. Symb. Comput., 25:5 (1998), 571–585 | DOI | MR | Zbl
[29] J. von zur Gathen, J. Gerhard, “Fast Algorithms for Taylor Shifts and Certain Difference Equations”, Proceedings of ISSAC' 97 (Maui, Hawaii, U.S.A., 1997), 40–47 | MR
[30] B. Kioustelidis, “Bounds for positive roots of polynomials”, J. Comput. Appl. Math., 16:2 (1986), 241–244 | DOI | MR | Zbl
[31] E. K. Lloyd, “On the forgotten Mr. Vincent”, Historia Mathematica, 6 (1979), 448–450 | DOI | MR
[32] Obreschkoff N., Zeros of Polynomials, Bulgarian Academic Monographs, 7, Sofia, 2003 | MR | Zbl
[33] A. M. Ostrowski, “Note on Vincent's Theorem”, The Annals of Mathematics (2), 52:3 (1950), 702–707 | DOI | MR | Zbl
[34] F. Rouillier, P. Zimmermann, “Efficient isolation of polynomial's real roots”, Journal of Computational and Applied Mathematics, 162 (2004), 33–50 | DOI | MR | Zbl
[35] J.-A. Serret, Cours d'Algèbre Supérieur, Vol. 1, 2, Gauthier–Villars, Paris, 1866; Copies of these volumes can be downloaded from http://www.archive.org/details/coursdalgebsuper01serrrich
[36] V. Sharma, Complexity of Real Root Isolation Using Continued Fractions, ISAAC07 preprint, 2007 | MR
[37] V. Sharma, Complexity Analysis of Algorithms in Algebraic Computation, Ph. D. Thesis, Department of Computer Sciences, Courant Institute of Mathematical Sciences, New York University, 2007
[38] D. Ştefănescu, “New bounds for positive roots of polynomials”, Journal of Universal Computer Science, 11:12 (2005), 2132–2141 | MR
[39] D. Ştefănescu, “Bounds for Real Roots and Applications to Orthogonal Polynomials”, Proceedings of the 10th International Workshop on Computer Algebra in Scientific Computing, CASC 2007 (Bonn, Germany, September 16–20, 2007), LNCS, 4770, eds. V. G. Ganzha, E. W. Mayr, E. V. Vorozhtsov, Springer Verlag, Berlin–Heidelberg, 377–391 | Zbl
[40] K. Thull, “Approximation by Continued Fraction of a Polynomial Real Root”, Proceedings of the 1984 ACM Symposium on Symbolic and Algebraic Computations, LNCS, 174, 1984, 367–377 | MR | Zbl
[41] E. P. Tsigaridas, I. Z. Emiris, “Univariate polynomial real root isolation: Continued fractions revisited”, ESA 2006, LNCS, 4168, eds. Y. Azar, T. Erlebach, 2006, 817–828 | MR | Zbl
[42] J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948
[43] A. Yu. Uteshev, Private Communication, September, 2007 | Zbl
[44] A. J. H. Vincent, “Sur la resolution des équations numériques”, Journal de Mathématiques Pures et Appliquées, 1 (1836), 341–372
[45] C. K. Yap, Fundamental Problems of Algorithmic Algebra, Oxford University Press, 2000 | MR | Zbl