TAILIEUCHUNG - Đề tài "Annals of Mathematics Lehmer’s problem for polynomials with odd coefficients "

We prove that if f (x) = n−1 ak xk is a polynomial with no cyclotomic k=0 factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k 1 + log 3 , 2n resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials | Annals of Mathematics Lehmer s problem for polynomials with odd coefficients By Peter Borwein Edward Dobrowolski and Michael J. Mossinghoff Annals of Mathematics 166 2007 347 366 Lehmer s problem for polynomials with odd coefficients By Peter Borwein Edward Dobrowolski and Michael J. Mossinghoff Abstract We prove that if f x Ak-0 akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak 1 mod 2 for 0 k n then Mahler s measure of f satisfies iogM f 5 1 -1 . n This resolves a problem of D. H. Lehmer 12 for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients degree n 1 and at least one noncyclotomic factor then at least one root a of f satisfies a 1 13- resolving a conjecture of Schinzel and Zassenhaus 21 for this class of polynomials. More generally we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials where each coefficient satisfies ak 1 mod m for a fixed integer m 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak 1 mod p for each k for a fixed prime p. Last we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point from both sides of Salem 19 numbers whose minimal polynomials have coefficients in 1 1 . 1. Introduction Mahler s measure of a polynomial f denoted M f is defined as the product of the absolute values of those roots of f that lie outside the unit disk multiplied by the absolute value of the leading coefficient. Writing f x The first author was supported in part by NSERC of Canada and MITACS. The authors thank the Banff International Research Station for hosting the workshop on The many aspects of Mahler s measure where this research began. 348 P. BORWEIN E. DOBROWOLSKI AND M. J. MOSSINGHOFF aH k 1 x ak we have d M f a ỊỊ max 1 ak . k 1 For f E Z x clearly M f 1 and by a classical theorem of Kronecker M f 1 precisely when f x is a product .

• Thạc sĩ - Tiến sĩ - Cao học
• Lehmer’s problem
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center