Polynomials

ISBN-13: 9780387406275 / Angielski / Miękka / 2003 / 455 str.

Edward J Barbeau

Opis

The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 "explorations" invite the reader to investigate research problems and related topics.

Recenzje

From the reviews:

E.J. Barbeau

Polynomials

"This book uses the medium of problems to enable us, the readers, to educate ourselves in matters polynomial. In each section we are led, after a brief introduction, into a sequence of problems on a certain topic. If we do these successfully, we find that we have mastered the basics of the topic. If we have any difficulties, we can refer first to the hints, and, failing these, to the detailed solutions. These form an important and substantial part of the book, and often refer the reader on to the research literature. The book, like good literature, can be read successfully at different levels, and would not be out of place in any mathematician's library."-MATHEMATICAL REVIEWS

"This is a two-faced book, and that's a good thing. One face is a set of enrichment materials for bright high school students. The other face is a fairly comprehensive textbook on algebraic properties of polynomials. ... The present book is an excellent introduction to the subject for anyone, from high schooler to professional." (Allen Stenger, The Mathematical Association of America, August, 2011)

Spis treści

1 Fundamentals.- 1.1 The Anatomy of a Polynomial of a Single Variable.- 1.1.5 Multiplication by detached coefficients.- 1.1.19 Even and odd polynomials.- E.1 Square of a polynomial.- E.2 Sets with equal polynomial-value sums.- E.3 Polynomials as generating functions.- 1.2 Quadratic Polynomials.- 1.2.1 Quadratic formula.- 1.2.4 Theory of the quadratic.- 1.2.14 Cauchy-Schwarz inequality.- 1.2.17 Arithmetic-geometric mean inequality.- 1.2.18 Approximation of quadratic irrational by a rational.- E.4 Graphical solution of the quadratic.- E.5 Polynomials, some of whose values are squares.- 1.3 Complex Numbers.- 1.3.8 De Moivre’s theorem.- 1.3.10 Square root of a complex number.- 1.3.15 Tchebychef polynomials.- E.6 Commuting polynomials.- 1.4 Equations of Low Degree.- 1.4.4 Cardan’s method for cubic.- 1.4.11 Descartes’ method for quartic.- 1.4.12 Ferrari’s method for quartic.- 1.4.13 Reciprocal equations.- E.7 The reciprocal equation substitution.- 1.5 Polynomials of Several Variables.- 1.5.2 Criterion for homogeneity.- 1.5.5 Elementary symmetric polynomials of 2 variables.- 1.5.8 Elementary symmetric polynomials of 3 variables.- 1.5.9 Arithmetic-geometric mean inequality for 3 numbers.- 1.5.10 Polynomials with n variables.- E.8 Polynomials in each variable separately.- E.9 The range of a polynomial.- E.10 Diophantine equations.- 1.6 Basic Number Theory and Modular Arithmetic.- 1.6.1 Euclidean algorithm.- 1.6.5 Modular arithmetic.- 1.6.6 Linear congruence.- E.11 Length of Euclidean algorithm.- E.12 The congruence a? ? b (mod m).- E.13 Polynomials with prime values.- E.14 Polynomials whose positive values are.- Fibonacci numbers.- 1.7 Rings and Fields.- 1.7.6 Zm.- E.15 Irreducible polynomials of low degree modulo p.- 1.8 Problems on Quadratics.- 1.9 Other Problems.- Hints.- 2 Evaluation, Division, and Expansion.- 2.1 Horner’s Method.- 2.1.8–9 Use of Horner’s method for Taylor expansion.- E.16 Number of multiplications for cn.- E.17 A Horner’s approach to the binomial expansion.- E.18 Factorial powers and summations.- 2.2 Division of Polynomials.- 2.2.2 Factor Theorem.- 2.2.4 Number of zeros cannot exceed degree of polynomial.- 2.2.7 Long division of polynomials; quotient and remainder.- 2.2.9 Division Theorem.- 2.2.12 Factor Theorem for two variables.- 2.2.15 Gauss’ Theorem on symmetric functions.- E.19 Chromatic polynomials.- E.20 The greatest common divisor of two polynomials.- E.21 The remainder for special polynomial divisors.- 2.3 The Derivative.- 2.3.4 Definition of derivative.- 2.3.5 Properties of the derivative.- 2.3.9 Taylor’s Theorem.- 2.3.15 Multiplicity of zeros.- E.22 Higher order derivatives of the composition of two functions.- E.23 Partial derivatives.- E.24 Homogeneous polynomials.- E.25 Cauchy-Riemann conditions.- E.26 The Legendre equation.- 2.4 Graphing Polynomials.- 2.4.6 Symmetry of cubic graph.- E.27 Intersection of graph of polynomial with lines.- E.28 Rolle’s Theorem.- 2.5 Problems.- Hints.- 3 Factors and Zeros.- 3.1 Irreducible Polynomials.- 3.1.3 Irreducibility of linear polynomials.- 3.1.10 Irreducibility over Q related to irreducibility over Z.- 3.1.12 Eisenstein criterion.- 3.2 Strategies for Factoring Polynomials over Z.- 3.2.6 Undetermined coefficients.- E.29 t2 ? t + a as a divisor of tn + 1 + b.- E.30 The sequence un(t).- 3.3 Finding Integer and Rational Roots: Newton’s Method of Divisors.- 3.3.5 Newton’s Method of Divisors.- E.31 Rational roots of nt2 + (n + 1)t ? (n + 2).- 3.4 Locating Integer Roots: Modular Arithmetic.- 3.4.7 Chinese Remainder Theorem.- E.32 Little Fermat Theorem.- E.33 Hensel’s Lemma.- 3.5 Roots of Unity.- 3.5.1 Roots of unity.- 3.5.7 Primitive roots of unity.- 3.5.9 Cyclotomic polynomials.- 3.5.18 Quadratic residue.- 3.5.19 Sicherman dice.- E.34 Degree of the cyclotomic polynomials.- E.35 Irreducibility of the cyclotomic polynomials.- E.36 Coefficients of the cyclotomic polynomials.- E.37 Little Fermat Theorem generalized.- 3.6 Rational Functions.- 3.6.4–6 Partial fractions.- E.38 Principal parts and residues.- 3.7 Problems on Factorization.- 3.8 Other Problems.- Hints.- 4 Equations.- 4.1 Simultaneous Equations in Two or Three Unknowns.- 4.1.2 Two linear homogeneous equations in three unknowns.- 4.1.7 Use of symmetric functions of zeros.- 4.2 Surd Equations.- 4.3 Solving Special Polynomial Equations.- 4.3.6 Surd conjugate.- 4.3.6–10 Field extensions.- E.39 Solving by radicals.- E.40 Constructions using ruler and compasses.- 4.4 The Fundamental Theorem of Algebra: Intersecting Curves.- 4.5 The Fundamental Theorem: Functions of a Complex Variable.- 4.6 Consequences of the Fundamental Theorem.- 4.6.1 Decomposition into linear factors.- 4.6.2 C as an algebraically closed field.- 4.6.3 Factorization over R.- 4.6.7 Uniqueness of factorization.- 4.6.8 Uniqueness of polynomial of degree n taking n + 1 assigned values.- 4.6.10 A criterion for irreducibility over Z.- E.41 Zeros of the derivative: Gauss-Lucas Theorem.- 4.7 Problems on Equations in One Variable.- 4.8 Problems on Systems of Equations.- 4.9 Other Problems.- Hints.- 5 Approximation and Location of Zeros.- 5.1 Approximation of Roots.- 5.1.1 The method of bisection.- 5.1.3 Linear interpolation.- 5.1.4 Horner’s Method.- 5.1.5 Newton’s Method.- 5.1.9 Successive approximation to a fixpoint.- E.42 Convergence of Newton approximations.- E.43 Newton’s Method according to Newton.- E.44 Newton’s Method and Hensel’s Lemma.- E.45 Continued fractions: Lagrange’s method of approximation.- E.46 Continued fractions: another approach for quadratics.- 5.2 Tests for Real Zeros.- 5.2.7 Descartes’ Rule of Signs.- 5.2.12 A bound on the real zeros.- 5.2.15 Rolle’s Theorem.- 5.2.17–20 Theorem of Fourier-Budan.- E.47 Proving the Fourier-Budan Theorem.- E.48 Sturm’s Theorem.- E.49 Oscillating populations.- 5.3 Location of Complex Roots.- 5.3.3 Cauchy’s estimate.- 5.3.8 Schur-Cohn criterion.- 5.3.9 Stable polynomials.- 5.3.10 Routh-Hurwitz criterion for a cubic.- 5.3.11 Nyquist diagram.- 5.3.13 Rouche’s Theorem.- E.50 Recursion relations.- 5.4 Problems.- Hints.- 6 Symmetric Functions of the Zeros.- 6.1 Interpreting the Coefficients of a Polynomial.- 6.1.9 Condition for real cubic to have real zeros.- 6.1.12 The zeros of a quartic expressed in terms of those of its resolvent sextic.- 6.2 The Discriminant.- 6.2.5 Discriminant of a cubic.- E.51 The discriminant of tn ? 1.- 6.3 Sums of the Powers of the Roots.- 6.3.6 The recursion formula.- E.52 Series approach for sum of powers of zeros.- E.53 A recursion relation.- E.54 Sum of the first nkth powers.- 6.4 Problems.- Hints.- 7 Approximations and Inequalities.- 7.1 Interpolation and Extrapolation.- 7.1.5 Lagrange polynomial.- 7.1.7–13 Finite differences.- 7.1.16 Factorial powers.- E.55 Building up a polynomial.- E.56 Propagation of error.- E.57 Summing by differences.- E.58 The absolute value function.- 7.2 Approximation on an Interval.- 7.2.1 Least squares.- 7.2.2 Alternation.- 7.2.5 Bernstein polynomials.- E.59 Taylor approximation.- E.60 Comparison of methods for approximating square roots.- 7.3 Inequalities.- 7.3.5–6 Generalizations of the AGM inequality.- 7.3.8 Bernoulli inequality.- 7.3.10 Newton’s inequalities.- E.61 AGM inequality for five variables.- 7.4 Problems on Inequalities.- 7.5 Other Problems.- Hints.- 8 Miscellaneous Problems.- E.62 Zeros of z?1[(1 + z)n ? 1 ? zn].- E.63 Two trigonometric products.- E.64 Polynomials all of whose derivatives have integer zeros.- E.65 Polynomials with equally spaced zeros.- E.66 Composition of polynomials of several variables.- E.67 The Mandelbrot set.- E.68 Sums of two squares.- E.69 Quaternions.- Hint.- Answers to Exercises and Solutions to Problems.- Notes on Explorations.- Further Reading.

O autorze

From the reviews:

"This book uses the medium of problems to enable us, the readers, to educate ourselves in matters polynomial. In each section we are led, after a brief introduction, into a sequence of problems on a certain topic. If we do these successfully, we find that we have mastered the basics of the topic ... I approached this book with some prejudice against its non-textbook approach; but it won me over entirely. It is not easy to write a book that will interest both the bright high-school student and the practicing mathematician. That it does so is a tribute to the author's scholarship, style, choice of material, and careful attention to detail." -Mathematical Reviews