This comprehensive book covers both longstanding results in the theory of polynomials and recent developments which have until now only been available in the research literature. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials, including those which are symmetric, integervalue or cyclotomic, and those of Chebyshev and Bernoulli. There follow chapters on Galois theory and ideals in polynomial rings. Finally there is a detailed discussion of Hilbert’s 17th problem on the representation of nonnegative polynomials as sums of squares of rational functions and generalizations.From the reviews:... Despite the appearance of this book in a series titled Algorithms and Computation of Mathematics, computation occupies only a small part of the monograph. It is best described as a useful reference for ones personal collection and a text for a fullyear course given to graduate or even senior undergraduate students. ..... the book under review is worth purchasing for the library and possibly even for ones own collection. The authors interest in the history and development of this area is evident, and we have pleasant glimpses of progress over the last three centuries. He exercises nice judgment in selection of arguments, with respect to both representativeness of approaches and elegance, so that the reader gains a synopsis of and guide to the literature, in which more detail can be found. ...E. Barbeau, SIAM Review 47, No. 3, 2005... the volume is packed with results and proofs that are well organised thematically into chapters and sections. What is unusual is to have a text that embraces and intermingles both analytic and algebraic aspects of the theory. Although the subject is about such basic objects, many tough results of considerable generality are incorporated and it is striking that refinements, both in theorems and proofs continued throughout the latter part of the Twentieth Century. ... There is a plentiful of problems, some of which might be challenging even for polynomial people; solutions to selected problems are also included.S.D.Cohen, MathSciNet, MR 2082772, 2005Problems concerning polynomials have impulsed resp. accompanied the development of algebra from its very beginning until today and over the centuries a lot of mathematical gems have been brought to light. This book presents a few of them, some being classical, but partly probably unknown even to experts, some being quite recently discovered. … Many historical comments and a clear style make the book very readable, so it can be recommended warmly to nonexperts already at an undergraduate level and, because of its contents, to experts as well.G.Kowol, Monatshefte für Mathematik 146, Issue 4, 2005
Algorithms and Computation in Mathematics • Volume 11 Editors Manuel Bronstein Arjeh M Cohen Henri Cohen David Eisenbud Bernd Sturmfels Victor V Prasolov Polynomials Translated from the Russian by Dimitry Leites 123 Victor V Prasolov Independent University of Moscow Department Mathematics Bolshoy Vlasievskij per.11 119002 Moscow, Russia e-mail: prasolov@mccme.ru Dimitry Leites (Translator) Stockholm University Department of Mathematics 106 91 Stockholm, Sweden e-mail: mleites@math.su.se Originally published by MCCME Moscow Center for Continuous Math Education in 2001 (Second Edition) Mathematics Subject Classification (2000): 12-XX, 12E05 Library of Congress Control Number: 2009935697 ISSN 1431-1550 ISBN 978-3-540-40714-0 (hardcover) ISBN 978-3-642-03979-9 (softcover) DOI 10.1007/978-3-642-03980-5 e-ISBN 978-3-642-03980-5 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004, First softcover printing 2010 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typeset by the translator Edited and reformatted by LE-TeX, Leipzig, using a Springer LATEX macro package Cover design: deblik, Berlin Printed on acid-free paper Preface The theory of polynomials constitutes an essential part of university courses of algebra and calculus Nevertheless, there are very few books entirely devoted to this theory.1 Though, after the first Russian edition of this book was printed, there appeared several books2 devoted to particular aspects of the polynomial theory, they have almost no intersection with this book The following classical references (not translated into Russian and therefore not mentioned in the Russian editions of this book) are rare exceptions: Barbeau E J., Polynomials Corrected reprint of the 1989 original Problem Books in Mathematics Springer-Verlag, New York, 1995 xxii+455 pp.; Borwein P., Erd´elyi T., Polynomials and polynomial inequalities Graduate Texts in Mathematics, 161 Springer-Verlag, New York, 1995 x+480 pp.; Obreschkoff N., Verteilung und Berechnung der Nullstellen reeller Polynome (German) VEB Deutscher Verlag der Wissenschaften, Berlin 1963 viii+298 pp For example, some recent ones: Macdonald I G., Affine Hecke algebras and orthogonal polynomials Cambridge Tracts in Mathematics, 157 Cambridge University Press, Cambridge, 2003 x+175 pp.; Phillips G M., Interpolation and approximation by polynomials CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, 14 Springer-Verlag, New York, 2003 xiv+312 pp.; Mason J C., Handscomb D C Chebyshev polynomials Chapman & Hall/CRC, Boca Raton, FL, 2003 xiv+341 pp.; Rahman Q I., Schmeisser G., Analytic theory of polynomials, London Math Soc Monographs (N.S.) 26, 2002; Sheil-Small T., Complex polynomials, Cambridge studies in adv math 75, 2002; Lomont J S., Brillhart J., Elliptic polynomials Chapman & Hall/CRC, Boca Raton, FL, 2001 xxiv+289 pp.; Krall A M., Hilbert space, boundary value problems and orthogonal polynomials Operator Theory: Advances and Applications, 133 Birkhă auser Verlag, Basel, 2002 xiv+352 pp.; Dunkl Ch F., Xu Yuan, Orthogonal polynomials of several variables Encyclopedia of Mathematics and its Applications, 81 Cambridge University Press, Cambridge, 2001 xvi+390 pp (Hereafter the translator’s footnotes.) VI Preface This book contains an exposition of the main results in the theory of polynomials, both classical and modern Considerable attention is given to Hilbert’s 17th problem on the representation of non-negative polynomials by the sums of squares of rational functions and its generalizations Galois theory is discussed primarily from the point of view of the theory of polynomials, not from that of the general theory of fields and their extensions More precisely: In Chapter we discuss, mostly classical, theorems about the distribution of the roots of a polynomial and of its derivative It is also shown how to determine the number of real roots to a real polynomial, and how to separate them Chapter deals with irreducibility criterions for polynomials with integer coefficients, and with algorithms for factorization of such polynomials and for polynomials with coefficients in the integers mod p In Chapter we introduce and study some special classes of polynomials: symmetric (polynomials which are invariant when the indeterminates are permuted), integer valued (polynomials which attain integer values at all integer points), cyclotomic (polynomials with all primitive nth roots of unity as roots), and some interesting classes introduced by Chebyshev, and by Bernoulli In Chapter we collect a lot of scattered results on properties of polynomials We discuss, e.g., how to construct polynomials with prescribed values in certain points (interpolation), how to represent a polynomial as a sum of powers of polynomials of degree one, and give a construction of numbers which are not roots of any polynomial with rational coefficients (transcendental numbers) Chapter is devoted to the classical Galois theory It is well known that the roots of a polynomial equation of degree at most four in one variable can be expressed in terms of radicals of arithmetic expressions of its coefficients A main application of Galois theory is that this is not possible in general for equations of degree five or higher In Chapter three classical Hilbert’s theorems are given: an ideal in a polynomial ring has a finite basis (Hilbert’s basis theorem); if a polynomial f vanishes on all common zeros of f1 , , fr , then some power of f is a linear combination (with polynomial coefficients) of f1 , , fr (Hilbert’s Nullstellensatz); and if M = ⊕Mi is a finitely generated module over a polynomial ring over K, then dimK Mi is a polynomial in i for large i (the Hilbert polynomial of M ) Furthermore, the theory of Grăobner bases is introduced Gră obner bases are a tool for calculations in polynomial rings An application is that solving systems of polynomial equations in several variables with finitely many solutions can be reduced to solving polynomial equations in one variable In the final Chapter considerable attention is given to Hilbert’s 17th problem on the representation of non-negative polynomials as the sum of squares of rational functions, and to its generalizations The Lenstra-LenstraLov` asz algorithm for factorization of polynomials with integer coefficients is discussed in an appendix Preface VII Two important results of the theory of polynomials whose exposition requires quite a lot of space did not enter the book: how to solve fifth degree equations by means of theta functions, and the classification of commuting polynomials These results are expounded in detail in two recently published books in which I directly participated: [Pr3] and [Pr4] During the work on this book I received financial support from the Russian Fund of Basic Research under Project No 01-01-00660 Acknowledgement Together with the translator, I am thankful to Dr Eastham for meticulous and friendly editing of the English and mathematics, to J Borcea, R Frăoberg, B Shapiro and V Kostov for useful comments V Prasolov Moscow, May 1999 VIII Preface Notational conventions As usual, Z denotes the set of all integers, N the subset of positive integers, Fp = Z/pZ for p prime (Z/nZ)∗ denotes the set of invertible elements of Z/nZ |S| denotes the cardinality of the set S R[x] denotes the ring of polynomials in one indeterminate x with coefficients in a commutative ring R [x] denotes the integer part of a given real number x, i.e., the greatest integer which is ≤ x Numbering of Theorems, Lemmas and Examples is usually continuous throughout each section, e.g., reference to Lemma 2.3.2 means that the Lemma is to be found in subsection 2.3 inside the same chapter Subsections are numbered separately, so Theorem 2.3.4 may occure in subsec 2.3.2 Certain Lemmas and Examples (considered of local importance) are numbered simply Lemma 1, and so on, and, to find it, the page is indicated in the reference Contents Roots of Polynomials 1.1 Inequalities for roots 1.1.1 The Fundamental Theorem of Algebra 1.1.2 Cauchy’s theorem 1.1.3 Laguerre’s theorem 1.1.4 Apolar polynomials 1.1.5 The Routh-Hurwitz problem 1.2 The roots of a given polynomial and of its derivative 1.2.1 The Gauss-Lucas theorem 1.2.2 The roots of the derivative and the focal points of an ellipse 1.2.3 Localization of the roots of the derivative 1.2.4 The Sendov-Ilieff conjecture 1.2.5 Polynomials whose roots coincide with the roots of their derivatives 1.3 The resultant and the discriminant 1.3.1 The resultant 1.3.2 The discriminant 1.3.3 Computing certain resultants and discriminants 1.4 Separation of roots 1.4.1 The Fourier–Budan theorem 1.4.2 Sturm’s Theorem 1.4.3 Sylvester’s theorem 1.4.4 Separation of complex roots 1.5 Lagrange’s series and estimates of the roots of a given polynomial 1.5.1 The Lagrange-Bă urmann series 1.5.2 Lagrange’s series and estimation of roots 1.6 Problems to Chapter 1.7 Solutions of selected problems 1 11 12 12 14 15 18 20 20 20 23 25 27 27 30 31 35 37 37 40 41 42 ... 12-XX, 12E05 Library of Congress Control Number: 2009935697 ISSN 143 1-1 550 ISBN 97 8-3 -5 4 0-4 071 4-0 (hardcover) ISBN 97 8-3 -6 4 2-0 397 9-9 (softcover) DOI 10.1007/97 8-3 -6 4 2-0 398 0-5 e-ISBN 97 8-3 -6 4 2-0 398 0-5 ... of integer-valued polynomials 85 3.2.2 Integer-valued polynomials in several variables 87 3.2.3 The q-analogue of integer-valued polynomials 88 3.3 The cyclotomic polynomials. .. complex variable V.V Prasolov, Polynomials, Algorithms and Computation in Mathematics 11, DOI 10.1007/97 8-3 -6 4 2-0 398 0-5 _1, © Springer-Verlag Berlin Heidelberg 2010 Roots of Polynomials Proof In