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Polynomials with Special Regard to Reducibility.

By: Series: Encyclopedia of Mathematics and its ApplicationsPublisher: Cambridge : Cambridge University Press, 2000Copyright date: ©2000Description: 1 online resource (570 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511151231
Subject(s): Genre/Form: Additional physical formats: Print version:: Polynomials with Special Regard to ReducibilityDDC classification:
  • 512.942
LOC classification:
  • QA161.P59 S337 2000
Online resources:
Contents:
Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- Introduction -- Notation -- 1 Arbitrary polynomials over an arbitrary field -- 1.1 Lüroth's theorem -- 1.2 Theorems of Gordan and E. Noether -- 1.3 Ritt's first theorem -- 1.4 Ritt's second theorem -- 1.5 Connection between reducibility and decomposability. The case of two variables -- 1.6 Kronecker's theorems on factorization of polynomials -- 1.7 Connection between reducibility and decomposability. The case of more than two variables -- 1.8 Some auxiliary results -- 1.9 A connection between irreducibility of a polynomial and of its substitution value after a specialization of some of the… -- 1.10 A polytope and a matrix associated with a polynomial -- 2 Lacunary polynomials over an arbitrary field -- 2.1 Theorems of Capelli and Kneser -- 2.2 Applications to polynomials in many variables -- 2.3 An extension of a theorem of Gourin -- 2.4 Reducibility of polynomials in many variables, that are trinomials with respect to one of them -- 2.5 Reducibility of quadrinomials in many variables -- 2.6 The number of terms of a power of a polynomial -- 3 Polynomials over an algebraically closed field -- 3.1 A theorem of E. Noether -- 3.2 Theorems of Ruppert -- 3.3 Salomon's and Bertini's theorems on reducibility -- 3.4 The Mahler measure of polynomials over C -- 4 Polynomials over a finitely generated field -- 4.1 A refinement of Gourin's theorem -- 4.2 A lower bound for the Mahler measure of a polynomial over Z -- 4.3 The greatest common divisor of… -- 4.4 Hilbert's irreducibility theorem -- 5 Polynomials over a number field -- 5.1 Introduction -- 5.2 The classes C(K, r, 1) -- 5.3 Families of diagonal ternary quadratic forms each isotropic over K -- 5.4 The class C(K, r, 2) -- 5.5 The class… -- 5.6 The class C(K, r, s) for arbitrary s.
5.7 The class C(K, r, s) for arbitrary s -- 5.8 The class C(K, r, s) for arbitrary s -- 5.9 A digression on kernels of lacunary polynomials -- 6 Polynomials over a Kroneckerian field -- 6.1 The Mahler measure of non-self-inversive polynomials -- 6.2 Non-self-inversive factors of a lacunary polynomial -- 6.3 Self-inversive factors of lacunary polynomials -- 6.4 The generalized Brauers-Hopf problem -- Appendices -- Appendix A. Algebraic functions of one variable -- Appendix B. Elimination theory -- Appendix C. Permutation groups and abstract groups -- Appendix D. Diophantine equations -- Appendix E. Matrices and lattices -- Appendix F. Finite fields and congruences -- Appendix G. Analysis -- Appendix I. Inequalities -- Appendix J. Distribution of primes -- Appendix K. Convexity -- Appendix by Umberto Zannier. Proof of Conjecture 1 -- 1. Tools from geometry -- 2. Lattices and algebraic groups -- 3. Weil heights -- 4. Heights in X… -- 5. Finiteness of maximal anomalous intersections -- Step 1. Increasing the dimension -- Step 2. Decreasing the dimension to 1 -- Step 3. Constructing anomalous intersection points -- Step 4. Application of Proposition 1 to conclude -- 6. Deduction of Conjecture 1 for number fields -- Bibliography -- Standard references -- References -- Index of definitions and conjectures -- Index of theorems -- Index of terms.
Summary: This book covers most of the known results on reducibility of polynomials.
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Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- Introduction -- Notation -- 1 Arbitrary polynomials over an arbitrary field -- 1.1 Lüroth's theorem -- 1.2 Theorems of Gordan and E. Noether -- 1.3 Ritt's first theorem -- 1.4 Ritt's second theorem -- 1.5 Connection between reducibility and decomposability. The case of two variables -- 1.6 Kronecker's theorems on factorization of polynomials -- 1.7 Connection between reducibility and decomposability. The case of more than two variables -- 1.8 Some auxiliary results -- 1.9 A connection between irreducibility of a polynomial and of its substitution value after a specialization of some of the… -- 1.10 A polytope and a matrix associated with a polynomial -- 2 Lacunary polynomials over an arbitrary field -- 2.1 Theorems of Capelli and Kneser -- 2.2 Applications to polynomials in many variables -- 2.3 An extension of a theorem of Gourin -- 2.4 Reducibility of polynomials in many variables, that are trinomials with respect to one of them -- 2.5 Reducibility of quadrinomials in many variables -- 2.6 The number of terms of a power of a polynomial -- 3 Polynomials over an algebraically closed field -- 3.1 A theorem of E. Noether -- 3.2 Theorems of Ruppert -- 3.3 Salomon's and Bertini's theorems on reducibility -- 3.4 The Mahler measure of polynomials over C -- 4 Polynomials over a finitely generated field -- 4.1 A refinement of Gourin's theorem -- 4.2 A lower bound for the Mahler measure of a polynomial over Z -- 4.3 The greatest common divisor of… -- 4.4 Hilbert's irreducibility theorem -- 5 Polynomials over a number field -- 5.1 Introduction -- 5.2 The classes C(K, r, 1) -- 5.3 Families of diagonal ternary quadratic forms each isotropic over K -- 5.4 The class C(K, r, 2) -- 5.5 The class… -- 5.6 The class C(K, r, s) for arbitrary s.

5.7 The class C(K, r, s) for arbitrary s -- 5.8 The class C(K, r, s) for arbitrary s -- 5.9 A digression on kernels of lacunary polynomials -- 6 Polynomials over a Kroneckerian field -- 6.1 The Mahler measure of non-self-inversive polynomials -- 6.2 Non-self-inversive factors of a lacunary polynomial -- 6.3 Self-inversive factors of lacunary polynomials -- 6.4 The generalized Brauers-Hopf problem -- Appendices -- Appendix A. Algebraic functions of one variable -- Appendix B. Elimination theory -- Appendix C. Permutation groups and abstract groups -- Appendix D. Diophantine equations -- Appendix E. Matrices and lattices -- Appendix F. Finite fields and congruences -- Appendix G. Analysis -- Appendix I. Inequalities -- Appendix J. Distribution of primes -- Appendix K. Convexity -- Appendix by Umberto Zannier. Proof of Conjecture 1 -- 1. Tools from geometry -- 2. Lattices and algebraic groups -- 3. Weil heights -- 4. Heights in X… -- 5. Finiteness of maximal anomalous intersections -- Step 1. Increasing the dimension -- Step 2. Decreasing the dimension to 1 -- Step 3. Constructing anomalous intersection points -- Step 4. Application of Proposition 1 to conclude -- 6. Deduction of Conjecture 1 for number fields -- Bibliography -- Standard references -- References -- Index of definitions and conjectures -- Index of theorems -- Index of terms.

This book covers most of the known results on reducibility of polynomials.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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