Lacunary Polynomials Over Finite Fields

Lacunary Polynomials Over Finite Fields focuses on reducible lacunary polynomials over finite fields, as well as stem polynomials, differential equations, and gaussian sums. The monograph first tackles preliminaries and formulation of Problems I, II, and III, including some basic concepts and notations, invariants of polynomials, stem polynomials, fully reducible polynomials, and polynomials with a restricted range. The text then takes a look at Problem I and reduction of Problem II to Problem III. Topics include reduction of the marginal case of Problem II to that of Problem III, proposition on power series, proposition on polynomials, and preliminary remarks on polynomial and differential equations. The publication ponders on Problem III and applications. Topics include homogeneous elementary symmetric systems of equations in finite fields; divisibility maximum properties of the gaussian sums and related questions; common representative systems of a finite abelian group with respect to given subgroups; and difference quotient of functions in finite fields. The monograph also reviews certain families of linear mappings in finite fields, appendix on the degenerate solutions of Problem II, a lemma on the greatest common divisor of polynomials with common gap, and two group-theoretical propositions. The text is a dependable reference for mathematicians and researchers interested in the study of reducible lacunary polynomials over finite fields.

Preface

Chapter I: Preliminaries and Formulation of Problems I, II, III

§ 1. Some Basic Concepts and Notations

§ 2. Invariants of Polynomials

§ 3. Polynomials with a Restricted Range

§ 4. Stem Polynomials

§ 5. Various Examples of Lacunary Polynomials

§ 6. Problems I, II, III Concerning Fully Reducible Lacunary Polynomials

§ 7. The λ-Differential Equation and the a, b-Polynomial Equation

§ 8. Fully Reducible Binomials

Chapter II: Problem I

§ 9. The Solution of Problem I

Chapter III: Reduction of Problem II to Problem III

§ 10. The solution of Problem II with the Exception of the Marginal Case and the Reduction of the Latter to the λ-Differential Equation

§ 11. Preliminary Remarks on Polynomial and Differential Equations

§ 12. Reduction of the λ-DifFerential Equation to the a, b-Polynomial Equation

§ 13. The Quadrature of the λ-Differential Equation

§ 14. The Reciprocal a, b-Polynomial Equation and the a, b-Power Series Equation

§ 15. Uniformization of the a, b-Power Series Equation

§ 16. A Proposition on Polynomials

§ 17. A Proposition on Power Series

§ 18. The Solution of the a, b-Polynomial Equation

§ 19. Reduction of the Marginal Case of Problem II to that of Problem III

Chapter IV: Problem III

§ 20. A Lemma on the Greatest Common Divisor of Polynomials with Common Gap

§ 21. The Theorem for Four P-Linear Polynomials

§ 22. Two Group-Theoretical Propositions

§ 23. Transformation of Problem III. A Necessary Condition for the Solutions

§ 24. Non-Primitive and Primitive Solutions of Problem III. Reduction of the Former

§ 25. The Regular Solutions of Problem III

§ 26. Explicit Determination of the Regular Solutions of Problem III

§ 27. Another Characterization of the Non-Primitive and Primitive Solutions of Problem III

§ 28. On the Primitive Solutions of Problem III

§ 29. The Solution of Problem III, Apart from the Marginal Case

§ 30. A Part of the Marginal Case of Problem III without Primitive Solutions

§ 31. A Further Part of the Marginal Case of Problem III without Primitive Solutions

Chapter V: The Solution of Problem II in Almost all Cases

§ 32. The Regular Solutions of Problem II

§ 33. Appendix on the Degenerate Solutions of Problem II

Chapter VI: Applications

§ 34. Certain Families of Linear Mappings in Finite Fields

§ 35. Application to Hajos's Theory

§ 36. On the Difference Quotient of Functions in Finite Fields

§ 37. Common Representative Systems of a Finite Abelian Group with Respect to Given Subgroups

§ 38. Divisibility Maximum Properties of Gaussian Sums and Related Questions

§ 39. Homogeneous Elementary Symmetric Systems of Equations in Finite Fields

Some Unsolved Problems

Literature

Subject Index

List of Theorems, Lemmas and Propositions