Polynomials with Special Regard to Reducibility (Encyclopedia of Mathematics and its Applications)

This treatise covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields, and finitely generated fields. The author includes several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields. Some of these results are based on the recent work of E. Bombieri and U. Zannier, presented here by Zannier in an appendix. The book also treats other subjects such as Ritt's theory of composition of polynomials, and properties of the Mahler measure and concludes with a bibliography of over 300 items. Cover......Page 1 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS......Page 2 Title: Polynomials with Special Regard to Reducibility......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 10 Acknowledgments......Page 11 Introduction......Page 12 Notation......Page 19 1.1 Lüroth’s theorem......Page 23 1.2 Theorems of Gordan and E. Noether......Page 26 1.3 Ritt’s first theorem......Page 29 1.4 Ritt’s second theorem......Page 35 1.5 Connection between reducibility and decomposability. The case of two variables......Page 63 1.6 Kronecker’s theorems on factorization of polynomials......Page 69 1.7 Connection between reducibility and decomposability. The case of more than two variables......Page 74 1.8 Some auxiliary results......Page 82 1.9 A connection between irreducibility of a polynomial and of its substitution value after a specialization of some of the variables......Page 86 1.10 A polytope and a matrix associated with a polynomial......Page 99 2.1 Theorems of Capelli and Kneser......Page 103 2.2 Applications to polynomials in many variables......Page 114 2.3 An extension of a theorem of Gourin......Page 121 2.4 Reducibility of polynomials in many variables, that are trinomials with respect to one of them......Page 133 2.5 Reducibility of quadrinomials in many variables......Page 178 2.6 The number of terms of a power of a polynomial......Page 197 3.1 A theorem of E. Noether......Page 212 3.2 Theorems of Ruppert......Page 215 3.3 Salomon’s and Bertini’s theorems on reducibility......Page 226 3.4 The Mahler measure of polynomials over C......Page 233 4.1 A refinement of Gourin’s theorem......Page 274 4.2 A lower bound for the Mahler measure of a polynomial over Z......Page 282 4.3 The greatest common divisor of KP(x^{n1}, . . . , x^{nk}) and KQ(x^{n1}, . . . , x^{nk})......Page 288 4.4 Hilbert’s irreducibility theorem......Page 309 5.1 Introduction......Page 326 5.2 The classes C_i(K, r, 1)......Page 330 5.3 Families of diagonal ternary quadratic forms each isotropic over K......Page 336 5.4 The class C_1(K, r, 2)......Page 342 5.5 The class C_i(K, r, 2) for i\neq 1......Page 350 5.6 The class C_0(K, r, s) for arbitrary s......Page 366 5.7 The class C_1(K, r, s) for arbitrary s......Page 380 5.8 The class C_2(K, r, s) for arbitrary s......Page 386 5.9 A digression on kernels of lacunary polynomials......Page 393 6.1 The Mahler measure of non-self-inversive polynomials......Page 401 6.2 Non-self-inversive factors of a lacunary polynomial......Page 431 6.3 Self-inversive factors of lacunary polynomials......Page 446 6.4 The generalized Brauers–Hopf problem......Page 484 Appendix A. Algebraic functions of one variable......Page 492 Appendix B. Elimination theory......Page 503 Appendix C. Permutation groups and abstract groups......Page 506 Appendix D. Diophantine equations......Page 509 Appendix E. Matrices and lattices......Page 510 Appendix F. Finite fields and congruences......Page 514 Appendix G. Analysis......Page 516 Appendix I. Inequalities......Page 519 Appendix J. Distribution of primes......Page 521 Appendix K. Convexity......Page 523 1. Tools from geometry......Page 528 2. Lattices and algebraic groups......Page 530 3. Weil heights......Page 533 4. Heights in X \cap H......Page 535 5. Finiteness of maximal anomalous intersections......Page 539 6. Deduction of Conjecture 1 for number fields......Page 549 Bibliography......Page 551 Index of definitions and conjectures......Page 566 Index of theorems......Page 567 Index of terms......Page 568 This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields. "The aim of this book is to cover most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix)." "The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure."--Jacket This unique book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Also included are results based on recent work of E. Bombieri and U. Zannier