Number theory and polynomials : [proceedings of a workshop on Number theory and polynomials held at Bristol University, 3-7 April 2006

Many Areas Of Active Research Within The Broad Field Of Number Theory Relate To Properties Of Polynomials, And This Volume Displays The Most Recent And Most Interesting Work On This Theme. The 2006 Number Theory And Polynomials Workshop In Bristol Drew Together International Researchers With A Variety Of Number-theoretic Interests, And The Book's Contents Reflect The Quality Of The Meeting. Topics Covered Include Recent Work On The Schur-siegel-smyth Trace Problem, Mahler Measure And Its Generalisations, The Merit Factor Problem, Barker Sequences, K3-surfaces, Self-inversive Polynomials, Newman's Inequality, Algorithms For Sparse Polynomials, The Integer Transfinite Diameter, Divisors Of Polynomials, Non-linear Recurrence Sequences, Polynomial Ergodic Averages, And The Hansen-mullen Primitivity Conjecture. With Surveys And Expository Articles Presenting The Latest Research, This Volume Is Essential For Graduates And Researchers Looking For A Snapshot Of Current Progress In Polynomials And Number Theory. The Trace Problem For Totally Positive Algebraic Integers / Julian Aguirre And Juan Carlos Peral ; Appendix / Jean-pierre Serre -- Mahler's Measure: From Number Theory To Geometry / Marie Jose Bertin -- Explicit Calculation Of Elliptic Fibrations Of K3-surfaces And Their Belyi-maps / Frits Beukers And Hans Montanus -- The Merit Factor Problem / Peter Borwein, Ron Ferguson And Joshua Knauer -- Barker Sequences And Flat Polynomials / Peter Borwein And Michael Mossinghoff -- The Hansen-mullen Primitivity Conjecture: Completion Of Proof / Stephen Cohen And Mateja Presern -- An Inequality For The Multiplicity Of The Roots Of A Polynomial / Arturas Dubickas -- Newman's Inequality For In Creasing Exponential Sums / Tamas Erdelyi -- On Primitive Divisors Of N² + B / Graham Everest And Glyn Harman. Irreducibility And Greatest Common Divisor Algorithms For Sparse Polynomials / Michael Filaseta, Andrew Granville And Andrzej Schinzel -- Consequences Of The Continuity Of The Monic Integer Transfinite Diameter / Jan Hilmar -- Nonlinear Recurrence Sequences And Laurent Polynomials / Andrew Hone -- Conjugate Algebraic Numbers On Conics : A Survey / James Mckee -- On Polynomial Ergodic Averages And Square Functions / Radhakrishnan Nair -- Polynomial Inequalities, Mahler's Measure, And Multipliers / Igor E. Pritsket -- Integer Transfinite Diameter And Computation Of Polynomials / Georges Rhin And Qiang Wu -- Smooth Divisors Of Polynomials / Eira Scourfield -- Self-inversive Polynomials With All Zeros On The Unit Circle / Christopher Sinclair And Jeffrey Vaaler -- The Mahler Measure Of Algebraic Numbers: A Survey / Chris Smyth. [edited By] James Mckee, Chris Smyth. This Volume Is The Proceedings Of A Workshop On 'number Theory And Polynomials' Held At Bristol University, 3-7 April 2006 ...--p. Vii. Includes Bibliographical References. Many Areas Of Active Research Within The Broad Field Of Number Theory Relate To Properties Of Polynomials, And This Volume Displays The Most Recent And Most Interesting Work On This Theme. The 2006 Number Theory And Polynomials Workshop In Bristol Drew Together International Researchers With A Variety Of Number-theoretic Interests, And The Book's Contents Reflect The Quality Of The Meeting. Topics Covered Include Recent Work On The Schur-siegel-smyth Trace Problem, Mahler Measure And Its Generalisations, The Merit Factor Problem, Barker Sequences, K3-surfaces, Self-inversive Polynomials, Newman's Inequality, Algorithms For Sparse Polynomials, The Integer Transfinite Diameter, Divisors Of Polynomials, Non-linear Recurrence Sequences, Polynomial Ergodic Averages, And The Hansen-mullen Primitivity Conjecture. With Surveys And Expository Articles Presenting The Latest Research, This Volume Is Essential For Graduates And Researchers Looking For A Snapshot Of Current Progress In Polynomials And Number Theory. The Trace Problem For Totally Positive Algebraic Integers / Julian Aguirre And Juan Carlos Peral ; Appendix / Jean-pierre Serre -- Mahler's Measure: From Number Theory To Geometry / Marie Jose Bertin -- Explicit Calculation Of Elliptic Fibrations Of K3-surfaces And Their Belyi-maps / Frits Beukers And Hans Montanus -- The Merit Factor Problem / Peter Borwein, Ron Ferguson And Joshua Knauer -- Barker Sequences And Flat Polynomials / Peter Borwein And Michael Mossinghoff -- The Hansen-mullen Primitivity Conjecture: Completion Of Proof / Stephen Cohen And Mateja Presern -- An Inequality For The Multiplicity Of The Roots Of A Polynomial / Arturas Dubickas -- Newman's Inequality For In Creasing Exponential Sums / Tamas Erdelyi -- On Primitive Divisors Of N2 + B / Graham Everest And Glyn Harman. Irreducibility And Greatest Common Divisor Algorithms For Sparse Polynomials / Michael Filaseta, Andrew Granville And Andrzej Schinzel -- Consequences Of The Continuity Of The Monic Integer Transfinite Diameter / Jan Hilmar -- Nonlinear Recurrence Sequences And Laurent Polynomials / Andrew Hone -- Conjugate Algebraic Numbers On Conics : A Survey / James Mckee -- On Polynomial Ergodic Averages And Square Functions / Radhakrishnan Nair -- Polynomial Inequalities, Mahler's Measure, And Multipliers / Igor E. Pritsket -- Integer Transfinite Diameter And Computation Of Polynomials / Georges Rhin And Qiang Wu -- Smooth Divisors Of Polynomials / Eira Scourfield -- Self-inversive Polynomials With All Zeros On The Unit Circle / Christopher Sinclair And Jeffrey Vaaler -- The Mahler Measure Of Algebraic Numbers: A Survey / Chris Smyth. [edited By] James Mckee, Chris Smyth. This Volume Is The Proceedings Of A Workshop On 'number Theory And Polynomials' Held At Bristol University, 3-7 April 2006 ...--p. Vii. Includes Bibliographical References. Cover......Page 1 Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 8 Index of authors......Page 9 List of participants......Page 12 Conference photograph, with key......Page 0 The trace problem for totally positive algebraic integers with an appendix by Jean-Pierre Serre......Page 16 Mahler's measure: from Number Theory to Geometry......Page 35 Explicit calculation of elliptic fibrations of K^3-surfaces and their Belyi-maps.......Page 48 The merit factor problem......Page 67 Barker sequences and flat polynomials.......Page 86 The Hansen-Mullen primitivity conjecture: completion of proof.......Page 104 An inequality for the multiplicity of the roots of a polynomial......Page 136 Newman's inequality for increasing exponential sums......Page 142 On primitive divisors of n^2 + b.......Page 157 Irreducibility and greatest common divisor algorithms for sparse polynomials.......Page 170 Consequences of the continuity of the monic integer transfinite diameter.......Page 192 Nonlinear recurrence sequences and Laurent polynomials.......Page 203 Conjugate algebraic numbers on conics: a survey......Page 226 On polynomial ergodic averages and square functions......Page 256 Polynomial inequalities, Mahler's measure, and multipliers......Page 270 Integer transfinite diameter and computation of polynomials.......Page 292 Smooth divisors of polynomials.......Page 301 Self-inversive polynomials with all zeros on the unit circle.......Page 327 The Mahler measure of algebraic numbers: a survey.......Page 337 Cambridge University Press Cover 1 Title Page 4 Copyright Page 5 Contents 6 Preface 8 Index of authors 9 List of participants 12 Conference photograph, with key -1 The trace problem for totally positive algebraic integers with an appendix by Jean-Pierre Serre 16 Mahler's measure: from Number Theory to Geometry 35 Explicit calculation of elliptic fibrations of K^3-surfaces and their Belyi-maps. 48 The merit factor problem 67 Barker sequences and flat polynomials. 86 The Hansen-Mullen primitivity conjecture: completion of proof. 104 An inequality for the multiplicity of the roots of a polynomial 136 Newman's inequality for increasing exponential sums 142 On primitive divisors of n^2 + b. 157 Irreducibility and greatest common divisor algorithms for sparse polynomials. 170 Consequences of the continuity of the monic integer transfinite diameter. 192 Nonlinear recurrence sequences and Laurent polynomials. 203 Conjugate algebraic numbers on conics: a survey 226 On polynomial ergodic averages and square functions 256 Polynomial inequalities, Mahler's measure, and multipliers 270 Integer transfinite diameter and computation of polynomials. 292 Smooth divisors of polynomials. 301 Self-inversive polynomials with all zeros on the unit circle. 327 The Mahler measure of algebraic numbers: a survey. 337 0521714672,9780521714679,9780511721274,0511721277 Title; Copyright; Contents; Preface; Index of authors; List of participants; Conference photograph, with key; The trace problem for totally positive algebraic integers; 1. Introduction; Acknowledgements; 2. Earlier results; 3. The method of auxiliary functions; 4. The trace problem and the integer Chebyshev problem; 5. Cyclotomic integers; Appendix A. The best lower estimate of the constant K; Appendix B. The letter from J.-P. Serre to C.J. Smyth; References; Mahler's measure: from Number Theory to Geometry; 1. Introduction; 2. Curves of genus 0; 3. Curves of genus 1; 4. Surfaces Containing the latest research, this volume is essential for readers looking for a snapshot of current progress in polynomials and number theory. Contributions by leading experts in the field include recent work on the Schur-Siegel-Smyth trace problem, Mahler measure, the merit factor problem, Barker sequences, and K3-surfaces.