Lacunary Polynomials Over Finite Fields

PrefaceChapter 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 BinomialsChapter II: Problem I § 9. The Solution of Problem IChapter 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 IIIChapter 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 SolutionsChapter 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 IIChapter 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 FieldsSome Unsolved ProblemsLiteratureSubject IndexList of Theorems, Lemmas and Propositions